A186686 - OEIS

%I #22 Nov 29 2017 15:13:57

%S 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,

%T 1,0,38,213,322,220,78,14,1,0,30,321,724,705,364,105,16,1,0,17,414,

%U 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

%N 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.

%H V. V. Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2011-2013.

%F 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).

%e 1,

%e 2,1,

%e 3,4,1,

%e 4,10,6,1,

%e 3,20,21,8,1,

%e 1,31,56,36,10,1,

%e 0,38,120,120,55,12,1,

%e 0,38,213,322,220,78,14,1,

%e 0,30,321,724,705,364,105,16,1

%p 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

%t 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}];

%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2017 *)

%K nonn,tabl,easy

%O 1,2

%A _Vladimir Kruchinin_, Feb 28 2011