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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. 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;

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.

Sequence in context: A015118 A340428 A174691 * A188644 A331899 A111830

Adjacent sequences: A156689 A156690 A156691 * A156693 A156694 A156695

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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Last modified February 2 19:34 EST 2023. Contains 360024 sequences. (Running on oeis4.)