A156727 -id:A156727

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A156692

Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(i+1) ) and m = 3, read by rows.

+10
8

1, 1, 1, 1, -7, 1, 1, 77, 77, 1, 1, -1155, 12705, -1155, 1, 1, 21945, 3620925, 3620925, 21945, 1, 1, -504735, 1582344225, -23735163375, 1582344225, -504735, 1, 1, 13627845, 982635763725, 280051192661625, 280051192661625, 982635763725, 13627845, 1

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

OFFSET

0,5

COMMENTS

Row sums are: {1, 2, -5, 156, 10397, 7285742, -20571484393, 562067684106392, 91653158600215578137, ...}.

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(i+1) ) and m = 3.

T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (3, 1, 1). - G. C. Greubel, Feb 25 2021

EXAMPLE

Triangle begins as:

1, 1;

1, -7, 1;

1, 77, 77, 1;

1, -1155, 12705, -1155, 1;

1, 21945, 3620925, 3620925, 21945, 1;

1, -504735, 1582344225, -23735163375, 1582344225, -504735, 1;

MATHEMATICA

(* First program *)

t[n_, k_]:= If[k==0, n!, Product[1 -(i+1)*(k+1), {j, n}, {i, 0, j-1}] ];

T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)

(* Second program *)

f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];

T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));

Table[T[n, k, 3, 1, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

PROG

(Sage)

@CachedFunction

def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))

flatten([[T(n, k, 3, 1, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 25 2021

(Magma)

f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;

[T(n, k, 3, 1, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A007318 (m=0), A156690 (m=1), A156691 (m=2), this sequence (m=3).

Cf. A156693, A156698, A156727.

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 13 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 25 2021

STATUS

approved

A156698

Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 3, read by rows.

+10
6

1, 1, 1, 1, -3, 1, 1, 33, 33, 1, 1, -627, 6897, -627, 1, 1, 16929, 3538161, 3538161, 16929, 1, 1, -592515, 3343562145, -63527680755, 3343562145, -592515, 1, 1, 25478145, 5032061028225, 2581447307479425, 2581447307479425, 5032061028225, 25478145, 1

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

OFFSET

0,5

COMMENTS

Row sums are: {1, 2, -1, 68, 5645, 7110182, -56841741493, 5172958787971592, 4953496772756652670937, ...}.

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(2*i-1) ) and m = 3.

T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (3, 2, -1). - G. C. Greubel, Feb 26 2021

EXAMPLE

Triangle begins as:

1, 1;

1, -3, 1;

1, 33, 33, 1;

1, -627, 6897, -627, 1;

1, 16929, 3538161, 3538161, 16929, 1;

1, -592515, 3343562145, -63527680755, 3343562145, -592515, 1;

MATHEMATICA

(* First program *)

t[n_, k_]:= If[k==0, n!, Product[1 -(2*i-1)*(k+1), {j, n}, {i, 0, j-1}] ];

T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 26 2021 *)

(* Second program *)

f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];

T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));

Table[T[n, k, 3, 2, -1], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 26 2021 *)

PROG

(Sage)

@CachedFunction

def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))

flatten([[T(n, k, 3, 2, -1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 26 2021

(Magma)

f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;

[T(n, k, 3, 2, -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 26 2021

CROSSREFS

Cf. A007318 (m=0), A156696 (m=1), A156697 (m=2), this sequence (m=3).

Cf. A156690, A156691, A156692, A156693.

Cf. A156692, A156699, A156727.

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 13 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 26 2021

STATUS

approved

A156722

Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 1, read by rows.

+10
6

1, 1, 1, 1, -1, 1, 1, 7, 7, 1, 1, -91, 637, -91, 1, 1, 1729, 157339, 157339, 1729, 1, 1, -43225, 74736025, -971568325, 74736025, -43225, 1, 1, 1339975, 57920419375, 14306343585625, 14306343585625, 57920419375, 1339975, 1, 1, -49579075, 66434721023125, -410234402317796875, 7794453644038140625, -410234402317796875, 66434721023125, -49579075, 1

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

OFFSET

0,8

COMMENTS

Row sums are: {1, 2, 1, 16, 457, 318138, -822182723, 28728530689952, 6974117708745434977, 19261978962188975367009202, ...}.

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 1.

T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (1, 3, -2). - G. C. Greubel, Feb 26 2021

EXAMPLE

Triangle begins as:

1, 1;

1, -1, 1;

1, 7, 7, 1;

1, -91, 637, -91, 1;

1, 1729, 157339, 157339, 1729, 1;

1, -43225, 74736025, -971568325, 74736025, -43225, 1;

1, 1339975, 57920419375, 14306343585625, 14306343585625, 57920419375, 1339975, 1;

MATHEMATICA

(* First program *)

t[n_, k_]:= If[k==0, n!, Product[1 -(3*i-2)*(k+1), {j, n}, {i, 0, j-1}] ];

T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 26 2021 *)

(* Second program *)

f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];

T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));

Table[T[n, k, 1, 3, -2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 26 2021 *)

PROG

(Sage)

@CachedFunction

def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))

flatten([[T(n, k, 1, 3, -2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 26 2021

(Magma)

f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;

[T(n, k, 1, 3, -2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 26 2021

CROSSREFS

Cf. A007318 (m=0), this sequence (m=1), A156725 (m=2), A156727 (m=3).

Cf. A156690, A156696, A156730.

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 14 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 26 2021

STATUS

approved

A156725

Triangle T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 2, read by rows.

+10
6

1, 1, 1, 1, -2, 1, 1, 22, 22, 1, 1, -440, 4840, -440, 1, 1, 12760, 2807200, 2807200, 12760, 1, 1, -484880, 3093534400, -61870688000, 3093534400, -484880, 1, 1, 22789360, 5525052438400, 3204530414272000, 3204530414272000, 5525052438400, 22789360, 1

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

OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 0, 46, 3962, 5639922, -55684588958, 6420110978999522, 8653645559546848833282, 120959123027642635275104364802, ...}.

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

T(n, k, m) = t(n,m)/( t(k,m) * t(n-k,m) ) with T(n, 0, m) = T(n, n, m) = 1, where t(n, m) = Product_{j=1..n} Product_{i=1..j-1} ( 1 - (m+1)*(3*i-2) ) and m = 2.

T(n, k, m, p, q) = (-p*(m+1))^(k*(n-k)) * (f(n,m,p,q)/(f(k,m,p,q)*f(n-k,m,p,q))) where Product_{j=1..n} Pochhammer( (q*(m+1) -1)/(p*(m+1)), j) for (m, p, q) = (2, 3, -2). - G. C. Greubel, Feb 26 2021

EXAMPLE

Triangle begins as:

1, 1;

1, -2, 1;

1, 22, 22, 1;

1, -440, 4840, -440, 1;

1, 12760, 2807200, 2807200, 12760, 1;

1, -484880, 3093534400, -61870688000, 3093534400, -484880, 1;

MATHEMATICA

(* First program *)

t[n_, k_]:= If[k==0, n!, Product[1 -(3*i-2)*(k+1), {j, n}, {i, 0, j-1}] ];

T[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 26 2021 *)

(* Second program *)

f[n_, m_, p_, q_]:= Product[Pochhammer[(q*(m+1) -1)/(p*(m+1)), j], {j, n}];

T[n_, k_, m_, p_, q_]:= (-p*(m+1))^(k*(n-k))*(f[n, m, p, q]/(f[k, m, p, q]*f[n-k, m, p, q]));

Table[T[n, k, 2, 3, -2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 26 2021 *)

PROG

(Sage)

@CachedFunction

def f(n, m, p, q): return product( rising_factorial( (q*(m+1)-1)/(p*(m+1)), j) for j in (1..n))

def T(n, k, m, p, q): return (-p*(m+1))^(k*(n-k))*(f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)))

flatten([[T(n, k, 2, 3, -2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 26 2021

(Magma)

f:= func< n, m, p, q | n eq 0 select 1 else m eq 0 select Factorial(n) else (&*[ 1 -(p*i+q)*(m+1): i in [0..j], j in [0..n-1]]) >;

T:= func< n, k, m, p, q | f(n, m, p, q)/(f(k, m, p, q)*f(n-k, m, p, q)) >;

[T(n, k, 2, 3, -2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 26 2021

CROSSREFS

Cf. A007318 (m=0), A156722 (m=1), this sequence (m=2), A156727 (m=3).

Cf. A156691, A156697, A156730.

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 14 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 26 2021

STATUS

approved

A156730

Triangle T(n, k) = Product_{j=1..k} Product_{i=0..j-1} ( 1 - (n-k+1)*(3*i-2) ) with T(n, 0) = 1 and T(n, n) = n!, read by rows.

+10
6

1, 1, 1, 1, 5, 2, 1, 7, -25, 6, 1, 9, -98, -875, 24, 1, 11, -243, -15092, 398125, 120, 1, 13, -484, -98415, 46483360, 3441790625, 720, 1, 15, -845, -404624, 1076168025, 4151893715200, -743856998828125, 5040, 1, 17, -1350, -1263275, 11501032576, 458947996781625, -14092191572383232000, -4983748910023583984375, 40320

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

OFFSET

0,5

COMMENTS

Row sums are: 1, 2, 8, -11, -939, 382922, 3488175820, -739704029345313, -4997840642636470626461, ...

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

Let the square array t(n, k) be given by t(n, k) = Product_{j=1..n} Product_{i=0..j-1} ( 1 - (k+1)*(3*i -2) ) with t(n, 0) = n!. The number triangle, T(n, k), is the downward antidiagonals, i.e. T(n, k) = t(k, n-k).

T(n, k) = (-3*(n-k+1))^binomial(k+1, 2)*Product_{j=1..k} Pochhammer( -(2*(n-k) + 3)/(3*(n-k+1)), j) with T(n, 0) = 1 and T(n, n) = n!. - G. C. Greubel, Feb 25 2021

EXAMPLE

Triangle begins as:

1, 1;

1, 5, 2;

1, 7, -25, 6;

1, 9, -98, -875, 24;

1, 11, -243, -15092, 398125, 120;

1, 13, -484, -98415, 46483360, 3441790625, 720;

1, 15, -845, -404624, 1076168025, 4151893715200, -743856998828125, 5040;

MATHEMATICA

(* First program *)

t[n_, k_]= If[k==0, n!, Product[1 -(3*i-2)*(k+1), {j, n}, {i, 0, j-1}]];

Table[t[k, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 25 2021 *)

(* Second program *)

T[n_, k_, p_, q_]:= If[k==0, 1, If[k==n, n!, (-p*(n-k+1))^Binomial[k+1, 2]*Product[ Pochhammer[(q*(n-k+1) -1)/(p*(n-k+1)), j], {j, k}]]];

Table[T[n, k, 3, -2], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 25 2021 *)

PROG

(Sage)

@CachedFunction

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

if (k==0): return 1

elif (k==n): return factorial(n)

else: return (-p*(n-k+1))^binomial(k+1, 2)*product( rising_factorial( (q*(n-k+1)-1)/(p*(n-k+1)), j) for j in (1..k))

flatten([[T(n, k, 3, -2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 24 2021

(Magma)

function T(n, k, p, q)

if k eq 0 then return 1;

elif k eq n then return Factorial(n);

else return (&*[1 - (n-k+1)*(p*m+q): m in [0..j-1], j in [1..k]]);

end if; return T;

end function;

[T(n, k, 3, -2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A156693, A156699, A156722, A156725, A156727.

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 14 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 25 2021

STATUS

approved

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