A186858 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).

0, 1, 2, 7, 30, 141, 703, 3653, 19567, 107270, 598955, 3394374, 19474061, 112884523, 660133281, 3889762894, 23071999207, 137648379106, 825454689096, 4972910455246, 30082871471563, 182660631479932, 1112852638627219, 6800915421684405, 41679320208737561, 256093061045766088

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1233 (terms n = 1..200 from Vincenzo Librandi)

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

Formula

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.

Maple

a:= n-> coeff(series(RootOf(x*(((((A+3)*A+4)*
        A+3)*A+2)*A+1)-A, A), x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 22 2018

Mathematica

terms = 25;
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}];
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jul 22 2018 *)

Prog

(PARI)
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);
vector(33, n, a(n-1)) /* show terms, Joerg Arndt, Mar 04 2011 */
(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

Crossrefs

Sequence in context: A353288 A299296 A116363 * A360102 A369441 A371432

Adjacent sequences: A186855 A186856 A186857 * A186859 A186860 A186861

Keyword

nonn

Author

Vladimir Kruchinin, Feb 28 2011

Extensions

a(0)=0 prepended by Alois P. Heinz, Jul 23 2018

Status

approved