A186686 Triangle T(n,k) of the coefficients [x^n] x^k*(x^5+3*x^4+4*x^3+3*x^2+2*x+1)^k, 1<=k<=n.

1, 2, 1, 3, 4, 1, 4, 10, 6, 1, 3, 20, 21, 8, 1, 1, 31, 56, 36, 10, 1, 0, 38, 120, 120, 55, 12, 1, 0, 38, 213, 322, 220, 78, 14, 1, 0, 30, 321, 724, 705, 364, 105, 16, 1, 0, 17, 414, 1400, 1897, 1353, 560, 136, 18, 1, 0, 6, 456, 2364, 4410, 4218, 2366, 816, 171, 20, 1, 0, 1, 427, 3515, 9020, 11374, 8365, 3860, 1140, 210, 22, 1

Offset

1,2

Links

Table of n, a(n) for n=1..78.

V. V. Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2011-2013.

Formula

T(n,k) = Sum_{s=k..n} binomial(s,n-s) * Sum_{j=0..k} binomial(k,j) * binomial(j,s-3*k+2*j).

Example

1,
2,1,
3,4,1,
4,10,6,1,
3,20,21,8,1,
1,31,56,36,10,1,
0,38,120,120,55,12,1,
0,38,213,322,220,78,14,1,
0,30,321,724,705,364,105,16,1

Maple

A186686 := proc(n, k) x*(1+2*x+3*x^2+4*x^3+3*x^4+x^5) ; expand(%^k) ; coeftayl(%, x=0, n) ; end proc: # R. J. Mathar, Mar 04 2011

Mathematica

T[n_, k_] := Sum[Binomial[s, n-s]*Sum[Binomial[k, j]*Binomial[j, s - 3*k + 2*j], {j, 0, k}], {s, k, n}];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 29 2017 *)

Crossrefs

Sequence in context: A204213 A143326 A327086 * A361045 A053122 A078812

Adjacent sequences: A186683 A186684 A186685 * A186687 A186688 A186689

Keyword

nonn,tabl,easy

Author

Vladimir Kruchinin, Feb 28 2011

Status

approved