A156697 -id:A156697

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A156691

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 = 2, read by rows.

+10
8

1, 1, 1, 1, -5, 1, 1, 40, 40, 1, 1, -440, 3520, -440, 1, 1, 6160, 542080, 542080, 6160, 1, 1, -104720, 129015040, -1419165440, 129015040, -104720, 1, 1, 2094400, 43865113600, 6755227494400, 6755227494400, 43865113600, 2094400, 1, 1, -48171200, 20177952256000, -52825879006208000, 739562306086912000, -52825879006208000, 20177952256000, -48171200, 1

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

OFFSET

0,5

COMMENTS

Row sums are: {1, 2, -3, 82, 2642, 1096482, -1161344798, 13598189404802, 633950903882665602, 301999235305843794118402, ...}.

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 = 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, 1, 1). - G. C. Greubel, Feb 25 2021

EXAMPLE

Triangle begins as:

1, 1;

1, -5, 1;

1, 40, 40, 1;

1, -440, 3520, -440, 1;

1, 6160, 542080, 542080, 6160, 1;

1, -104720, 129015040, -1419165440, 129015040, -104720, 1;

1, 2094400, 43865113600, 6755227494400, 6755227494400, 43865113600, 2094400, 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, 2], {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, 2, 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, 2, 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, 2, 1, 1): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A007318 (m=0), A156690 (m=1), this sequence (m=2), A156692 (m=3).

Cf. A156693, A156697, A156725.

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

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

A156696

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 = 1, read by rows.

+10
5

1, 1, 1, 1, -1, 1, 1, 5, 5, 1, 1, -45, 225, -45, 1, 1, 585, 26325, 26325, 585, 1, 1, -9945, 5817825, -52360425, 5817825, -9945, 1, 1, 208845, 2076963525, 243004732425, 243004732425, 2076963525, 208845, 1, 1, -5221125, 1090405850625, -2168817236893125, 28194624079610625, -2168817236893125, 1090405850625, -5221125, 1

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

OFFSET

0,8

COMMENTS

Row sums are: {1, 2, 1, 12, 137, 53822, -40744663, 490163809592, 23859170407083377, 14660989220762621919002, -54998077449004520067705092623, ...}.

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 = 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, 2, -1). - G. C. Greubel, Feb 25 2021

EXAMPLE

Triangle begins as:

1, 1;

1, -1, 1;

1, 5, 5, 1;

1, -45, 225, -45, 1;

1, 585, 26325, 26325, 585, 1;

1, -9945, 5817825, -52360425, 5817825, -9945, 1;

1, 208845, 2076963525, 243004732425, 243004732425, 2076963525, 208845, 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, 1], {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, 1, 2, -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, 1, 2, -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, 1, 2, -1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 25 2021

CROSSREFS

Cf. A007318 (m=0), this sequence (m=1), A156697 (m=2), A156698 (m=3).

cf. A156690, A156691, A156692, A156693.

Cf. A156690, A156699, A156722.

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Feb 13 2009

EXTENSIONS

Edited by G. C. Greubel, Feb 25 2021

STATUS

approved

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