A256890 -id:A256890

A257610

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.

+10
14

1, 2, 2, 4, 20, 4, 8, 132, 132, 8, 16, 748, 2112, 748, 16, 32, 3964, 25124, 25124, 3964, 32, 64, 20364, 256488, 552728, 256488, 20364, 64, 128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128, 256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.

Sum_{k=0..n} T(n, k) = A007559(n).

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 2. - G. C. Greubel, Mar 20 2022

EXAMPLE

Triangle begins as:

1;

2, 2;

4, 20, 4;

8, 132, 132, 8;

16, 748, 2112, 748, 16;

32, 3964, 25124, 25124, 3964, 32;

64, 20364, 256488, 552728, 256488, 20364, 64;

128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128;

256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;

MATHEMATICA

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];

Table[T[n, k, 3, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

PROG

(Sage)

def T(n, k, a, b): # A257610

if (k<0 or k>n): return 0

elif (n==0): return 1

else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)

flatten([[T(n, k, 3, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

CROSSREFS

Cf. A007559 (row sums), A038208, A142458, A257620, A257622, A257624, A257626.

Cf. A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619.

See similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 03 2015

STATUS

approved

A257609

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.

+10
13

1, 2, 2, 4, 16, 4, 8, 88, 88, 8, 16, 416, 1056, 416, 16, 32, 1824, 9664, 9664, 1824, 32, 64, 7680, 76224, 154624, 76224, 7680, 64, 128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128, 256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.

Sum_{k=0..n} T(n, k) = A002866(n).

From G. C. Greubel, Mar 21 2022: (Start)

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 2, and b = 2.

T(n, n-k) = T(n, k).

T(n, 0) = A000079(n).

T(n, 1) = 2*A100575(n+1). (End)

EXAMPLE

Triangle begins as:

1;

2, 2;

4, 16, 4;

8, 88, 88, 8;

16, 416, 1056, 416, 16;

32, 1824, 9664, 9664, 1824, 32;

64, 7680, 76224, 154624, 76224, 7680, 64;

128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128;

256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;

MATHEMATICA

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];

Table[T[n, k, 2, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

PROG

(Magma)

function T(n, k, a, b)

if k lt 0 or k gt n then return 0;

elif k eq 0 or k eq n then return 1;

else return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b);

end if; return T;

end function;

[T(n, k, 2, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022

(Sage)

def T(n, k, a, b): # A257609

if (k<0 or k>n): return 0

elif (n==0): return 1

else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)

flatten([[T(n, k, 2, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

CROSSREFS

Cf. A000079, A002866 (row sums), A060187, A100575, A257611, A257613, A257615.

Cf. A038208, A256890, A257610, A257612, A257614, A257616, A257617, A257618, A257619.

Cf. similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 03 2015

STATUS

approved

A257620

Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.

+10
12

1, 3, 3, 9, 36, 9, 27, 297, 297, 27, 81, 2106, 5346, 2106, 81, 243, 13851, 73386, 73386, 13851, 243, 729, 87480, 868239, 1761264, 868239, 87480, 729, 2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187, 6561, 3293622, 95843088, 578903274, 1024762590, 578903274, 95843088, 3293622, 6561

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.

Sum_{k=0..n} T(n, k) = A034001(n).

From G. C. Greubel, Feb 28 2022: (Start)

t(k, n) = t(n, k).

T(n, n-k) = T(n, k).

t(0, n) = T(n, 0) = A000244(n). (End)

EXAMPLE

Array t(n,k) begins as:

1, 3, 9, 27, 81, 243, ...;

3, 36, 297, 2106, 13851, 87480, ...;

9, 297, 5346, 73386, 868239, 9388791, ...;

27, 2106, 73386, 1761264, 34158753, 578903274, ...;

81, 13851, 868239, 34158753, 1024762590, 25791697782, ...;

243, 87480, 9388791, 578903274, 25791697782, 928501120152, ...;

729, 540189, 95843088, 8959544136, 575025893586, 28788563928042, ...;

Triangle T(n,k) begins as:

1;

3, 3;

9, 36, 9;

27, 297, 297, 27;

81, 2106, 5346, 2106, 81;

243, 13851, 73386, 73386, 13851, 243;

729, 87480, 868239, 1761264, 868239, 87480, 729;

2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187;

MATHEMATICA

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1, k, p, q] + (p*n+q)*t[n, k-1, p, q]]];

T[n_, k_, p_, q_]= t[n-k, k, p, q];

Table[T[n, k, 3, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 28 2022 *)

PROG

(Sage)

@CachedFunction

def t(n, k, p, q):

if (n<0 or k<0): return 0

elif (n==0 and k==0): return 1

else: return (p*k+q)*t(n-1, k, p, q) + (p*n+q)*t(n, k-1, p, q)

def A257620(n, k): return t(n-k, k, 3, 3)

flatten([[A257620(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

CROSSREFS

Cf. A000244, A034001 (row sums), A142458, A257610, A257622, A257624, A257626.

Cf. A038221, A257180, A257611, A257621, A257623, A257625, A257627.

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 09 2015

STATUS

approved

A257611

Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.

+10
11

1, 3, 3, 9, 30, 9, 27, 213, 213, 27, 81, 1308, 2982, 1308, 81, 243, 7431, 32646, 32646, 7431, 243, 729, 40314, 310263, 587628, 310263, 40314, 729, 2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187, 6561, 1099704, 22059036, 113360904, 191433990, 113360904, 22059036, 1099704, 6561

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 2*n + 3.

Sum_{k=0..n} T(n, k) = A051578(n).

From G. C. Greubel, Feb 28 2022: (Start)

t(k, n) = t(n, k).

T(n, n-k) = T(n, k).

t(0, n) = T(n, 0) = A000244(n). (End)

EXAMPLE

Array t(n,k) begins as:

1, 3, 9, 27, 81, 243, ...;

3, 30, 213, 1308, 7431, 40314, ...;

9, 213, 2982, 32646, 310263, 2695923, ...;

27, 1308, 32646, 587628, 8701545, 113360904, ...;

81, 7431, 310263, 8701545, 191433990, 3579465642, ...;

243, 40314, 2695923, 113360904, 3579465642, 93066106692, ...;

729, 212505, 22059036, 1351133676, 59641127202, 2104476295026, ...;

Triangle T(n,k) begins as:

1;

3, 3;

9, 30, 9;

27, 213, 213, 27;

81, 1308, 2982, 1308, 81;

243, 7431, 32646, 32646, 7431, 243;

729, 40314, 310263, 587628, 310263, 40314, 729;

2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187;

MATHEMATICA

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1, k, p, q] + (p*n+q)*t[n, k-1, p, q]]];

T[n_, k_, p_, q_]= t[n-k, k, p, q];

Table[T[n, k, 2, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 28 2022 *)

PROG

(PARI) f(x) = 2*x + 3;

T(n, k) = t(n-k, k);

t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1, m) + f(n)*t(n, m-1)));

tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); \\ Michel Marcus, May 06 2015

(Sage)

@CachedFunction

def t(n, k, p, q):

if (n<0 or k<0): return 0

elif (n==0 and k==0): return 1

else: return (p*k+q)*t(n-1, k, p, q) + (p*n+q)*t(n, k-1, p, q)

def A257611(n, k): return t(n-k, k, 2, 3)

flatten([[A257611(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

CROSSREFS

Cf. A000244, A051578 (row sums), A060187, A257609, A257613, A257615.

Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 06 2015

STATUS

approved

A257180

Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.

+10
10

1, 3, 3, 9, 24, 9, 27, 141, 141, 27, 81, 726, 1410, 726, 81, 243, 3471, 11406, 11406, 3471, 243, 729, 15828, 81327, 136872, 81327, 15828, 729, 2187, 69873, 533259, 1390521, 1390521, 533259, 69873, 2187, 6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561, 19683, 1277619, 19489380, 105311556, 237144642, 237144642, 105311556, 19489380, 1277619, 19683

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.

Sum_{k=0..n} T(n, k) = A001725(n+5).

From G. C. Greubel, Feb 22 2022: (Start)

t(k, n) = t(n, k).

T(n, n-k) = T(n, k).

t(0, n) = T(n, 0) = A000244(n). (End)

EXAMPLE

Array t(n,k) begins as:

1, 3, 9, 27, 81, 243, ... A000244;

3, 24, 141, 726, 3471, 15828, ...;

9, 141, 1410, 11406, 81327, 533259, ...;

27, 726, 11406, 136872, 1390521, 12609198, ...;

81, 3471, 81327, 1390521, 19467294, 237144642, ...;

243, 15828, 533259, 12609198, 237144642, 3794314272, ...;

729, 69873, 3295152, 105311556, 2607816498, 53824862658, ...;

Triangle T(n,k) begins as:

1;

3, 3;

9, 24, 9;

27, 141, 141, 27;

81, 726, 1410, 726, 81;

243, 3471, 11406, 11406, 3471, 243;

729, 15828, 81327, 136872, 81327, 15828, 729;

2187, 69873, 533259, 1390521, 1390521, 533259, 69873, 2187;

6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561;

MATHEMATICA

f[n_]:= n+3;

t[n_, k_]:= t[n, k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1, k] +f[n]*t[n, k-1]]];

T[n_, k_]= t[n-k, k];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)

PROG

(PARI) f(x) = x + 3;

T(n, k) = t(n-k, k);

t(n, m) = {if (!n && !m, return(1)); if (n < 0 || m < 0, return (0)); f(m)*t(n-1, m) + f(n)*t(n, m-1); }

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); } \\ Michel Marcus, Apr 23 2015

(Sage)

def f(n): return n+3

@CachedFunction

def t(n, k):

if (n<0 or k<0): return 0

elif (n==0 and k==0): return 1

else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)

def A257627(n, k): return t(n-k, k)

flatten([[A257627(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2022

CROSSREFS

Cf. A000244, A008292, A001725 (row sums), A038221, A256890, A257606.

Cf. A257607, A257611, A257620, A257621, A257623, A257625, A257627.

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, Apr 17 2015

STATUS

approved

A257612

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.

+10
10

1, 2, 2, 4, 24, 4, 8, 184, 184, 8, 16, 1216, 3680, 1216, 16, 32, 7584, 53824, 53824, 7584, 32, 64, 46208, 674752, 1507072, 674752, 46208, 64, 128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128, 256, 1677312, 84892672, 636233728, 1196803584, 636233728, 84892672, 1677312, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Corresponding entries in this triangle and in A060187 differ only by powers of 2. - F. Chapoton, Nov 04 2020

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.

Sum_{k=0..n} T(n,k) = A047053(n).

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 4, and b = 2. - G. C. Greubel, Mar 20 2022

EXAMPLE

Triangle begins as:

1;

2, 2;

4, 24, 4;

8, 184, 184, 8;

16, 1216, 3680, 1216, 16;

32, 7584, 53824, 53824, 7584, 32;

64, 46208, 674752, 1507072, 674752, 46208, 64;

128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;

MATHEMATICA

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];

Table[T[n, k, 4, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

PROG

(PARI) f(x) = 4*x + 2;

T(n, k) = t(n-k, k);

t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1, m) + f(n)*t(n, m-1)));

tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); \\ Michel Marcus, May 06 2015

(Sage)

def T(n, k, a, b): # A257612

if (k<0 or k>n): return 0

elif (n==0): return 1

else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)

flatten([[T(n, k, 4, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

CROSSREFS

Cf. A047053 (row sums), A060187, A142459, A257621.

Cf. A038208, A256890, A257609, A257610, A257614, A257616, A257617, A257618, A257619.

See similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 06 2015

STATUS

approved

A257614

Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.

+10
10

1, 2, 2, 4, 28, 4, 8, 244, 244, 8, 16, 1844, 5856, 1844, 16, 32, 13260, 101620, 101620, 13260, 32, 64, 93684, 1511160, 3455080, 1511160, 93684, 64, 128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128, 256, 4609588, 269011408, 2121603436, 4047202720, 2121603436, 269011408, 4609588, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.

Sum_{k=0..n} T(n, k) = A008546(n).

From G. C. Greubel, Mar 01 2022: (Start)

t(k, n) = t(n, k).

T(n, n-k) = T(n, k).

t(0, n) = T(n, 0) = A000079(n). (End)

EXAMPLE

Array t(n,k) begins as:

1, 2, 4, 8, 16, ... A000079;

2, 28, 244, 1844, 13260, ...;

4, 244, 5856, 101620, 1511160, ...;

8, 1844, 101620, 3455080, 91981880, ...;

16, 13260, 1511160, 91981880, 4047202720, ...;

32, 93684, 20663388, 2121603436, 146321752612, ...;

64, 657836, 269011408, 44675623468, 4648698508440, ...;

Triangle T(n,k) begins as:

1;

2, 2;

4, 28, 4;

8, 244, 244, 8;

16, 1844, 5856, 1844, 16;

32, 13260, 101620, 101620, 13260, 32;

64, 93684, 1511160, 3455080, 1511160, 93684, 64;

128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128;

MATHEMATICA

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1, k, p, q] + (p*n+q)*t[n, k-1, p, q]]];

T[n_, k_, p_, q_]= t[n-k, k, p, q];

Table[T[n, k, 5, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)

PROG

(Sage)

@CachedFunction

def t(n, k, p, q):

if (n<0 or k<0): return 0

elif (n==0 and k==0): return 1

else: return (p*k+q)*t(n-1, k, p, q) + (p*n+q)*t(n, k-1, p, q)

def A257614(n, k): return t(n-k, k, 5, 2)

flatten([[A257614(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022

CROSSREFS

Cf. A000079, A008546 (row sums), A142460, A257623.

Cf. A038208, A256890, A257609, A257610, A257612, A257616, A257617, A257618, A257619.

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 09 2015

STATUS

approved

A257616

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.

+10
10

1, 2, 2, 4, 32, 4, 8, 312, 312, 8, 16, 2656, 8736, 2656, 16, 32, 21664, 175424, 175424, 21664, 32, 64, 174336, 3019200, 7016960, 3019200, 174336, 64, 128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128, 256, 11182592, 722956288, 5907889664, 11379596800, 5907889664, 722956288, 11182592, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.

Sum_{k=0..n} T(n, k) = A049308(n).

From G. C. Greubel, Mar 21 2022: (Start)

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 6, and b = 2.

T(n, n-k) = T(n, k).

T(n, 0) = A000079(n).

T(n, 1) = (2^n/3)*(2^(2*n+1) - (3*n+2)). (End)

EXAMPLE

Triangle begins as:

1;

2, 2;

4, 32, 4;

8, 312, 312, 8;

16, 2656, 8736, 2656, 16;

32, 21664, 175424, 175424, 21664, 32;

64, 174336, 3019200, 7016960, 3019200, 174336, 64;

128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128;

MATHEMATICA

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];

Table[T[n, k, 6, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

PROG

(Sage)

def T(n, k, a, b): # A257610

if (k<0 or k>n): return 0

elif (n==0): return 1

else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)

flatten([[T(n, k, 6, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

CROSSREFS

Cf. A000079, A049308 (row sums), A142461, A257625.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257617, A257618, A257619.

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 09 2015

STATUS

approved

A257617

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.

+10
10

1, 2, 2, 4, 36, 4, 8, 388, 388, 8, 16, 3676, 12416, 3676, 16, 32, 33564, 283204, 283204, 33564, 32, 64, 303260, 5538184, 13027384, 5538184, 303260, 64, 128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 128

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.

Sum_{k=0..n} T(n, k) = A144827(n).

From G. C. Greubel, Mar 24 2022: (Start)

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 7, and b = 2.

T(n, n-k) = T(n, k).

T(n, 0) = A000079(n).

T(n, 1) = (4*9^n - 2^n*(7*n + 4))/7.

T(n, 2) = (2^(n-1)*(49*n^2 +7*n -12) + 11*2^(4*n+1) - 4*(7*n+4)*9^n)/49. (End)

EXAMPLE

1;

2, 2;

4, 36, 4;

8, 388, 388, 8;

16, 3676, 12416, 3676, 16;

32, 33564, 283204, 283204, 33564, 32;

64, 303260, 5538184, 13027384, 5538184, 303260, 64;

128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 128;

MATHEMATICA

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];

Table[T[n, k, 7, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

PROG

(Sage)

def T(n, k, a, b): # A257617

if (k<0 or k>n): return 0

elif (n==0): return 1

else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)

flatten([[T(n, k, 7, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

CROSSREFS

Cf. A000079, A142462, A144827 (row sums), A257627.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257618, A257619

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 09 2015

STATUS

approved

A257619

Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.

+10
10

1, 2, 2, 4, 44, 4, 8, 564, 564, 8, 16, 6436, 22560, 6436, 16, 32, 71404, 637844, 637844, 71404, 32, 64, 786948, 15470232, 36994952, 15470232, 786948, 64, 128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.

Sum_{k=0..n} T(n, k) = A144829(n).

From G. C. Greubel, Mar 24 2022: (Start)

T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 9, and b = 2.

T(n, n-k) = T(n, k).

T(n, 0) = A000079(n).

T(n, 1) = (1/9)*(4*11^n - 2^n*(9*n + 4)).

T(n, 2) = (1/81)*(26*20^n - 4*(4+9*n)*11^n - 2^(n-1)*(20 + 9*n - 81*n^2)). (End)

EXAMPLE

Triangle begins as:

1;

2, 2;

4, 44, 4;

8, 564, 564, 8;

16, 6436, 22560, 6436, 16;

32, 71404, 637844, 637844, 71404, 32;

64, 786948, 15470232, 36994952, 15470232, 786948, 64;

128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128;

MATHEMATICA

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];

Table[T[n, k, 9, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

PROG

(PARI) f(x) = 9*x + 2;

t(n, m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, f(m)*t(n-1, m) + f(n)*t(n, m-1)));

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", "); ); print(); ); } \\ Michel Marcus, May 23 2015

(Sage)

def T(n, k, a, b): # A257619

if (k<0 or k>n): return 0

elif (n==0): return 1

else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)

flatten([[T(n, k, 9, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

CROSSREFS

Cf. A000079, A144829 (row sums), A257608.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618.

Similar sequences listed in A256890.

KEYWORD

nonn,tabl

AUTHOR

Dale Gerdemann, May 09 2015

STATUS

approved