A173611 Self-convolution of A173610.

1, 2, 5, 12, 32, 86, 226, 588, 1596, 4372, 12065, 33344, 91344, 249584, 677896, 1836048, 5039672, 13877256, 38405640, 106482832, 296373672, 826270666, 2307068226, 6445895588, 17963996648, 50028938140, 139149397474

Offset

0,2

Links

Table of n, a(n) for n=0..26.

Formula

G.f. satisfies: A(x) = [C(x^2) + x*B(x^2)]^2 where

B(x) = Sum_{n>=0} a(n)^2*x^n = g.f. of A173612 and

C(x) = 1 + Sum_{n>=0} a(n)*a(n+1)*x^(n+1) = g.f. of A173613.

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 86*x^5 +...
Describe the g.f. of A173612 by:
B(x) = 1 + 4*x + 25*x^2 + 144*x^3 + 1024*x^4 +...+ a(n)^2*x^n +...
and describe the g.f. of A173613 by:
C(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 384*x^4 +...+ a(n)*a(n+1)*x^n +...
then the g.f. of this sequence is given by:
A(x) = [C(x^2) + x*B(x^2)]^2
where A(x) is the square of the g.f. of A173610:
A(x)^(1/2) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 60*x^6 +...

Prog

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

Crossrefs

Cf. A173610, A173612, A173613.

Sequence in context: A039809 A335456 A345200 * A292211 A293348 A188287

Adjacent sequences: A173608 A173609 A173610 * A173612 A173613 A173614

Keyword

nonn

Author

Paul D. Hanna, Feb 22 2010

Extensions

Edited by Paul D. Hanna, Feb 23 2010

Status

approved