A186858 - OEIS

%I #32 Sep 08 2022 08:45:56

%S 0,1,2,7,30,141,703,3653,19567,107270,598955,3394374,19474061,

%T 112884523,660133281,3889762894,23071999207,137648379106,825454689096,

%U 4972910455246,30082871471563,182660631479932,1112852638627219,6800915421684405,41679320208737561,256093061045766088

%N G.f. A(x) satisfies A(x) = x*(A(x)^5 + 3*A(x)^4 + 4*A(x)^3 + 3*A(x)^2 + 2*A(x) + 1).

%H Alois P. Heinz, <a href="/A186858/b186858.txt">Table of n, a(n) for n = 0..1233</a> (terms n = 1..200 from Vincenzo Librandi)

%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 a(n) = sum(k=0..n-1, binomial(k+n,n-k-1)*sum(j=0..n, binomial(n,j)*binomial(j,k-2*n+2*j)))/n, n>0.

%p a:= n-> coeff(series(RootOf(x*(((((A+3)*A+4)*

%p         A+3)*A+2)*A+1)-A, A), x, n+1), x, n):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 22 2018

%t terms = 25;

%t A[_] = 1; Do[A[x_] = x*(A[x]^5 + 3A[x]^4 + 4A[x]^3 + 3A[x]^2 + 2A[x] + 1) + O[x]^(terms+1), {terms+1}];

%t CoefficientList[A[x], x] // Rest (* _Jean-François Alcover_, Jul 22 2018 *)

%o (PARI)

%o a(n)=sum(k=0,n, binomial(k+n+1,n-k)*sum(j=0,n+1, binomial(n+1,j)*binomial(j,k-2*n-2+2*j)))/(n+1);

%o vector(33,n,a(n-1)) /* show terms, _Joerg Arndt_, Mar 04 2011 */

%o (Magma) [&+[Binomial(k+n,n-k-1)*(Binomial(n,i)*Binomial(i,k-2*n+2*i))/n: k in [0..25], i in [0..n]]: n in [1..25]]; // _Vincenzo Librandi_, Jul 23 2018

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Feb 28 2011

%E a(0)=0 prepended by _Alois P. Heinz_, Jul 23 2018