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A174109 Triangle read by rows: T(n, k) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, q) = Product_{j=1..n} (j+q)!/(j-1)! and q = 8. 2
1, 1, 1, 1, 10, 1, 1, 55, 55, 1, 1, 220, 1210, 220, 1, 1, 715, 15730, 15730, 715, 1, 1, 2002, 143143, 572572, 143143, 2002, 1, 1, 5005, 1002001, 13026013, 13026013, 1002001, 5005, 1, 1, 11440, 5725720, 208416208, 677352676, 208416208, 5725720, 11440, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 12, 112, 1652, 32892, 862864, 28066040, 1105659414, 51177188350, 2734044648194, ...}.

These sequences (q >= 2) are a generalization of A056939.

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n, q) = Product_{j=1..n} (j+q)!/(j-1)! and q=8.

T(n, k, q) = (G(q+2)*G(k+1)*G(n+q+2)*G(n-k+1))/(G(n+1)*G(k+q+2)*G(n-k+q+ 2)), where G(x) is the Barnes G-function (see A000178). - G. C. Greubel, Apr 13 2019

EXAMPLE

Triangle begins as:

  1.

  1,     1.

  1,    10,       1.

  1,    55,      55,         1.

  1,   220,    1210,       220,         1.

  1,   715,   15730,     15730,       715,         1.

  1,  2002,  143143,    572572,    143143,      2002,       1.

  1,  5005, 1002001,  13026013,  13026013,   1002001,    5005,     1.

  1, 11440, 5725720, 208416208, 677352676, 208416208, 5725720, 11440, 1.

MATHEMATICA

c[n_, q_]:= Product[i+j, {j, 0, q}, {i, 1, n}];

T[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);

Table[T[n, k, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 13 2019 *)

T[n_, k_, q_]:= (BarnesG[n+q+2]*BarnesG[k+1]*BarnesG[n-k+1]*BarnesG[q+2] )/(BarnesG[n-k+q+2]*BarnesG[k+q+2]*BarnesG[n+1]);

Table[T[n, k, 8], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 13 2019 *)

PROG

(PARI) {c(m, q) = prod(j=1, m, (j+q)!/(j-1)!)};

for(n=0, 10, for(k=0, n, print1(c(n, 8)/(c(k, 8)*c(n-k, 8)), ", "))) \\ G. C. Greubel, Apr 13 2019

CROSSREFS

Cf. A056939 (q=2), A056940 (q=3), A056941 (q=4), A142465 (q=5), A142467 (q=6), A142468 (q=7), this sequence (q=8).

Sequence in context: A157277 A157629 A154336 * A171692 A152971 A142459

Adjacent sequences:  A174106 A174107 A174108 * A174110 A174111 A174112

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Mar 08 2010

EXTENSIONS

Edited by G. C. Greubel, Apr 13 2019

STATUS

approved

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Last modified February 27 01:52 EST 2020. Contains 332299 sequences. (Running on oeis4.)