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A156766 Triangle T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = Product_{j=1..n} ( j!*((k+1)^j -1)/k ), f(n, 0) = n!, and m = 2, read by rows. 3
1, 1, 1, 1, 8, 1, 1, 78, 78, 1, 1, 960, 9360, 960, 1, 1, 14520, 1742400, 1742400, 14520, 1, 1, 262080, 475675200, 5854464000, 475675200, 262080, 1, 1, 5508720, 180465667200, 33594378048000, 33594378048000, 180465667200, 5508720, 1, 1, 132249600, 91065752064000, 305980926935040000, 4627961519892480000, 305980926935040000, 91065752064000, 132249600, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

FORMULA

T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = Product_{j=1..n} ( j!*((k+1)^j -1)/k ), f(n, 0) = n!, and m = 2.

T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), where T(0, k, m) = 1, f(n, k) = (-1/k)^n * BarnesG(n+2) * q-Pochhammer(k+1, k+1, n) and m = 2. - G. C. Greubel, Jun 19 2021

EXAMPLE

Triangle begins as:

1;

1, 1;

1, 8, 1;

1, 78, 78, 1;

1, 960, 9360, 960, 1;

1, 14520, 1742400, 1742400, 14520, 1;

1, 262080, 475675200, 5854464000, 475675200, 262080, 1;

MATHEMATICA

(* First program *)

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

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

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 19 2021 *)

(* Second program *)

f[n_, k_]:= If[k==0, n!, (-1)^n*BarnesG[n+2] QPochhammer[k+1, k+1, n]/k^n];

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

Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)

PROG

(Magma)

f:= func< n, k | n eq 0 select 1 else k eq 0 select Factorial(n) else (&*[Factorial(j)*((k+1)^j-1): j in [1..n]])/k^n >;

T:= func< n, k, m | n eq 0 select 1 else f(n, m)/(f(k, m)*f(n-k, m)) >;

[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021

(Sage)

from sage.combinat.q_analogues import q_pochhammer

@CachedFunction

def f(n, k): return factorial(n) if (k==0) else (-1/k)^n*product( factorial(j) for j in (1..n) )*q_pochhammer(n, k+1, k+1)

def T(n, k, m): return 1 if (n==0) else f(n, m)/(f(k, m)*f(n-k, m))

flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

CROSSREFS

Cf. A007318 (m=0), A156765 (m=1), this sequence (m=2), A156767 (m=3).

Sequence in context: A178048 A174728 A015121 * A178046 A176340 A111835

Adjacent sequences: A156763 A156764 A156765 * A156767 A156768 A156769

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 15 2009

EXTENSIONS

Edited by G. C. Greubel, Jun 19 2021

STATUS

approved

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Last modified January 30 15:50 EST 2023. Contains 359945 sequences. (Running on oeis4.)