A253256 G.f. satisfies: A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.

1, 2, 11, 79, 647, 5727, 53367, 515802, 5123303, 51977485, 536320688, 5610909773, 59379328267, 634538481389, 6837466955193, 74210071037031, 810527496757335, 8901979424068377, 98253966680382102, 1089260346498608721, 12123804391067414676, 135427509933882292680, 1517725698030921469890

Offset: 0

Author

Paul D. Hanna, May 31 2015

Keywords

• nonn

Comments

Self-convolution yields A253255.

Examples

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 79*x^3 + 647*x^4 + 5727*x^5 + 53367*x^6 +...
where A(x) = (1 - x^3*A(x)^6) / (1 - x*A(x)^2)^2.
The logarithm begins:
log(A(x)) = 2*x + 18*x^2 + 179*x^3 + 1874*x^4 + 20202*x^5 + 221943*x^6 + 2470827*x^7/7 +...+ A168595(n)/2*x^n/n +...
		

Crossrefs

Cf. A168595, A106228, A064641, A253255.

Programs

PARI

{a(n) = local(A=1); A = sqrt( (1/x)*serreverse( x*(1-x)^4/(1-x^3)^2 +x^2*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

PARI

{A168595(n) = sum(k=0, 2*n, binomial(2*n, k) * polcoeff((1+x+x^2)^n, k) )}
{a(n) = local(A=1); A = exp( sum(k=1,n+1, A168595(k)/2 * x^k/k) +x*O(x^n)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = exp( Sum_{n>=1} A168595(n)/2 * x^n/n ), where A168595(n) = Sum_{k=0..2*n} binomial(n,k)*trinomial(n,k).

(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x)^4/(1-x^3)^2 ) ).

(3) A(x) = sqrt( (1-x*A(x) - sqrt(1 - 6*x*A(x) - 3*x^2*A(x)^2)) / (2*x*(1+x*A(x))) ).