A216116 G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x^4*A(x)).

1, 1, 1, 1, 2, 4, 7, 11, 17, 28, 49, 87, 152, 262, 453, 794, 1408, 2507, 4462, 7943, 14179, 25415, 45713, 82398, 148731, 268859, 486890, 883411, 1605582, 2922259, 5325377, 9716564, 17750332, 32464980, 59443403, 108951953, 199886003, 367052947, 674620772, 1240963218

Offset: 0

Author

Paul D. Hanna, Oct 29 2012

Keywords

• nonn

Examples

			A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 + 17*x^8 + 28*x^9 +...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^3)*x +
(1 + 2^2*x^3 + x^6)*x^2/2 +
(1 + 3^2*x^3 + 3^2*x^6 + x^9)*x^3/3 +
(1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)*x^4/4 +
(1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15)*x^5/5 +
(1 + 6^2*x^3 + 15^2*x^6 + 20^2*x^9 + 15^2*x^12 + 6^2*x^15 + x^18)*x^6/6 +...
		

Crossrefs

Cf. A023427, A023432, A004148.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x^4*A)+x*O(x^n)); polcoeff(A, n)}

PARI

{a(n)=polcoeff(exp(sum(m=1, n+1, sum(j=0, m, binomial(m, j)^2*(x+x*O(x^n))^(3*j))*x^m/m)), n)}

PARI

{a(n)=polcoeff(((1-x-x^4) - sqrt((1-x-x^4)^2 - 4*x^5 +x^6*O(x^n)))/(2*x^5), n)}
for(n=0,45,print1(a(n),", "))

Formula

G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(3*k) ).

G.f.: A(x) = ((1-x-x^4) - sqrt((1-x-x^4)^2 - 4*x^5))/(2*x^5).

a(n) = A023427(n+1) for n>=0.