A304121 Expansion of A(x) = 1 + x + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3.

1, 2, 3, 8, 24, 73, 230, 751, 2512, 8549, 29520, 103192, 364441, 1298336, 4660273, 16837743, 61187249, 223489715, 820040293, 3021286338, 11172619317, 41454901924, 154285693649, 575826677260, 2154643706124, 8081453618583, 30377809192246, 114421431917805

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = Sum_{i=0..n} Sum_{k=0..n-i} Sum_{j=0..k} C(i,j)*C(j,k-j)*C(k+1,n-k-i)* C(k+i,i)/(k+1).

Maple

a:= n-> coeff(series(RootOf((A*x)^3+(A*x)^2+
            A*(x-1)+x+1, A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 14 2018

Mathematica

Array[Sum[Sum[Sum[Binomial[i, j] Binomial[j, k - j], {j, 0, k}] Binomial[k + 1, # - k - i] Binomial[k + i, i]/(k + 1), {k, 0, # - i}], {i, 0, # + 1}] &, 31, 0] (* Michael De Vlieger, May 10 2018 *)
n = 27; A = Sum[a[k] x^k, {k, 0, n}] + x O[x]^n; Table[a[k], {k, 0, n}] /.
Solve[LogicalExpand[1 + x + x A + x^2 A^2 + x^3 A^3 == A]] // Flatten (* Peter Luschny, May 14 2018 *)

Prog

(Maxima)
a(n):=sum(sum(((sum(binomial(i, j)*binomial(j, k-j), j, 0, k))*binomial(k+1, n-k-i)* binomial(k+i, i))/(k+1), k, 0, n-i), i, 0, n);

Crossrefs

Sequence in context: A271269 A331874 A089847 * A038561 A055981 A182212

Adjacent sequences: A304118 A304119 A304120 * A304122 A304123 A304124

Keyword

nonn

Author

Vladimir Kruchinin, May 07 2018

Status

approved