A178658 Triangle T(n,k) read by rows: the coefficient [x^k] of the series (1-x)^(2n-1)*Sum_{l>=0} A001263(n+3*l,3*l+1)*x^l, in row n>=1 with exponents k>=0.

1, 1, 7, 1, 1, 45, 96, 20, 1, 168, 1316, 1730, 420, 10, 1, 481, 9486, 35959, 35959, 9486, 481, 1, 1, 1165, 48103, 395692, 974863, 816025, 226765, 17248, 196, 1, 2507, 193830, 2977338, 14467635, 26621385, 19654740, 5661084, 558168, 13496, 28, 1, 4935

Offset

1,3

Comments

Row sums are 1, 9, 162, 3645, 91854, 2480058, 70150212, 2051893701, 61556811030,

1883638417518,....

Links

Table of n, a(n) for n=1..44.

Formula

Let A(n,m) = A001263(n+m,m+1), then T(n,k) = [x^k] (1-x)^(2n-1) * Sum_{l>=0} A(n,3l) *x^l.

If n == 2 (mod 3), then T(n,k) = T(n,5(n-2)/3+2-k).

Example

1;
1, 7, 1;
1, 45, 96, 20;
1, 168, 1316, 1730, 420, 10;
1, 481, 9486, 35959, 35959, 9486, 481, 1;
1, 1165, 48103, 395692, 974863, 816025, 226765, 17248, 196;
1, 2507, 193830, 2977338, 14467635, 26621385, 19654740, 5661084, 558168, 13496, 28;
1, 4935, 662019, 17287475, 146146455, 494277453, 735137025, 494277453, 146146455, 17287475, 662019, 4935, 1;
1, 9058, 1993006, 82927275, 1124706060, 6284964498, 16252722579, 20538756768, 12843595065, 3874383980, 524761304, 27575207, 415404, 825;

Mathematica

p[x_, n_] = (1 - x)^(2*n - 1)*Sum[(Binomial[3*k + n, 3* k] Binomial[3*k + n, 1 + 3*k]/(3*k + n))*x^k, {k, 0, Infinity}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]

Crossrefs

Cf. A001263, A034839

Sequence in context: A166973 A157156 A022170 * A156602 A203389 A174689

Adjacent sequences: A178655 A178656 A178657 * A178659 A178660 A178661

Keyword

nonn,tabf

Author

Roger L. Bagula, Jun 01 2010

Extensions

Edited by R. J. Mathar, May 13 2016

Status

approved