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A128084 Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1. 67
1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 5, 7, 8, 8, 7, 5, 3, 1, 1, 4, 9, 16, 24, 32, 39, 44, 46, 44, 39, 32, 24, 16, 9, 4, 1, 1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 1, 1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919, 2254, 2565, 2832, 3037, 3166, 3210, 3166, 3037, 2832, 2565, 2254, 1919, 1580, 1255, 958, 699, 484, 315, 190, 104, 50, 20, 6, 1, 1, 7, 27, 77, 181, 371, 686, 1170, 1869, 2827, 4082, 5662, 7581, 9835, 12399, 15225, 18242, 21358, 24464, 27440, 30162, 32510, 34376, 35672, 36336 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

See A128080 for the triangle of coefficients of q in the q-analog of the odd double factorials.

LINKS

Paul D. Hanna, Rows n=0..30 of triangle, in flattened form.

Eric Weisstein's World of Mathematics, q-Factorial from MathWorld.

A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.

A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.

A. V. Yurkin, New binomial and new view on light theory, LAP Lambert Academic Publishing, 2013, 78 pages.

EXAMPLE

The row sums form A000165, the even double factorial numbers:

[1, 2, 8, 48, 384, 3840, 46080, 645120, ..., (2n)!!, ...].

Triangle begins:

1;

1, 1;

1, 2,  2,  2,  1;

1, 3,  5,  7,  8,  8,   7,   5,   3,   1;

1, 4,  9, 16, 24, 32,  39,  44,  46,  44,  39,  32,  24,  16,   9,   4,   1;

1, 5, 14, 30, 54, 86, 125, 169, 215, 259, 297, 325, 340, 340, 325, 297, 259, 215, 169, 125, 86, 54, 30, 14, 5, 1;

1, 6, 20, 50, 104, 190, 315, 484, 699, 958, 1255, 1580, 1919, 2254, 2565, 2832, 3037, 3166, 3210, 3166, 3037, 2832, 2565, 2254, 1919, 1580, 1255, 958, 699, 484, 315, 190, 104, 50, 20, 6, 1;

1, 7, 27, 77, 181, 371, 686, 1170, 1869, 2827, 4082, 5662, 7581, 9835, 12399, 15225, 18242, 21358, 24464, 27440, 30162, 32510, 34376, 35672, 36336, 36336, ...; ...

MATHEMATICA

t[n_, k_] := If[k < 0 || k > n^2, 0, If[n == 0, 1, Coefficient[ Series[ Product[ (1 - q^(2*j))/(1 - q), {j, 1, n}], {q, 0, n^2}], q, k]]]; Table[t[n, k], {n, 0, 6}, {k, 0, n^2}] // Flatten (* Jean-Fran├žois Alcover, Mar 06 2013, translated from Pari *)

PROG

(PARI) {T(n, k) = if(k<0||k>n^2, 0, if(n==0, 1, polcoeff( prod(j=1, n, (1-q^(2*j))/(1-q)), k, q) ))}

for(n=0, 8, for(k=0, n^2, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A000165 ((2n)!!); A128085 (central terms); A128086 (diagonal), A128087 (row squared sums); A128080, A002522 (row lengths).

The growth series for the affine Coxeter groups B_2 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084.

Sequence in context: A152067 A286756 A193884 * A131823 A089722 A172356

Adjacent sequences:  A128081 A128082 A128083 * A128085 A128086 A128087

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Feb 14 2007

STATUS

approved

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Last modified February 22 02:13 EST 2019. Contains 320381 sequences. (Running on oeis4.)