A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2*n) / Product_{k=1..n} (1 - x^k*A(x))^2.

1, 1, 4, 18, 91, 489, 2751, 15985, 95218, 578324, 3568084, 22299964, 140885754, 898292262, 5772951668, 37355908797, 243184468271, 1591567315702, 10465836784159, 69114490893596, 458171948148640, 3047865264442504, 20339282134624054, 136122586785459512

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

G.f. satisfies the identities:

(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n+1) / Product_{k=1..n} (1 - x^k*A(x)).

(2) A(x) = 1/(1 - Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1 - x^k*A(x)) ).

Example

G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 91*x^4 + 489*x^5 + 2751*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + x*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^4/((1-x*A(x))^2*(1-x^2*A(x))^2) + x^9*A(x)^6/((1-x*A(x))^2*(1-x^2*A(x))^2*(1-x^3*A(x))^2) +...
(1) A(x) = 1 + x*A(x)^2/(1-x*A(x)) + x^2*A(x)^3/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)^4/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, 1-x^k*A+x*O(x^n))^2)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(m+1)/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A196150, A145268, A206637.

Sequence in context: A367724 A355247 A269450 * A367875 A172964 A317135

Adjacent sequences: A206636 A206637 A206638 * A206640 A206641 A206642

Keyword

nonn

Author

Paul D. Hanna, Feb 11 2012

Status

approved