A143927 G.f. satisfies: A(x) = (1 + x*A(x) + x^2*A(x)^2)^2.

1, 2, 7, 28, 123, 572, 2769, 13806, 70414, 365636, 1926505, 10273870, 55349155, 300783420, 1646828655, 9075674700, 50304255210, 280248358964, 1568399676946, 8813424968192, 49709017472751, 281306750922072, 1596802663432503

Offset: 0

Author

Paul D. Hanna, Sep 08 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A006605, A027908, A143926.

Programs

Mathematica

Table[GegenbauerC[n,-2n-2,-1/2]/(n+1),{n,0,12}] (* Emanuele Munarini, Oct 20 2016 *)
n = 20;
A = Sum[a[k] x^k, {k, 0, n}] + x O[x]^n;
Table[a[k], {k, 0, n}] /. Reverse[Solve[LogicalExpand[(1 + x A + x^2 A^2)^2 == A]]] (* Emanuele Munarini, Oct 20 2016 *)

Maxima

makelist(ultraspherical(n,-2*n-2,-1/2)/(n+1),n,0,12); /* Emanuele Munarini, Oct 20 2016 */

PARI

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=(1+x*A+x^2*A^2)^2);polcoeff(A,n)}

Formula

Self-convolution of A006605.

Bisection of A143926.

a(n) = ((24*n+12)*A006605(n) + (3*n+5)*A006605(n+1))/(13*n+17). - Mark van Hoeij, Jul 14 2010

a(n) = (1/(n+1))*Sum_{j=0..2*n+2} (binomial(j,2*j-3*n-4)*binomial(2*n+2 ,j)). - Vladimir Kruchinin, Dec 24 2010

a(n) = GegenbauerPoly(n,-2n-2,-1/2)/(n+1). - Emanuele Munarini, Oct 20 2016

a(n) = T(2*n+2, n)/(n+1), where T(n,k) = A027907(n,k). - Emanuele Munarini, Oct 20 2016

The g.f. A(x) satisfies 1 + x*A'(x)/A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 266*x^3 + ..., the g.f. of A027908. - Peter Bala, Aug 03 2023