A199409 G.f. satisfies: A(x) = Sum_{n>=0} A(x)^n * x^(n^2) * (1 - x^(2*n+1))/(1 - x).

1, 1, 2, 4, 8, 17, 37, 82, 184, 417, 954, 2200, 5109, 11937, 28040, 66179, 156857, 373205, 891034, 2134072, 5125944, 12344835, 29802478, 72109852, 174839832, 424742526, 1033697149, 2519947080, 6152807700, 15045156972, 36840289213, 90326900587, 221741403579, 544982530105

Offset

0,3

Links

Table of n, a(n) for n=0..33.

Formula

Define f(z,q) = Sum_{n>=0} z^n * q^(n^2) then g.f. A(q) satisfies:

A(q) = (f(A(q),q) - q*f(q^2*A(q),q))/(1-q).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 37*x^6 + 82*x^7 +...
where the g.f. A(x) satisfies the equivalent expressions:
A(x) = 1 + A(x)*x*(1-x^3)/(1-x) + A(x)^2*x^4*(1-x^5)/(1-x) + A(x)^3*x^9*(1-x^7)/(1-x) + A(x)^4*x^16*(1-x^9)/(1-x) +...
A(x) = 1 + A(x)*(x + x^2 + x^3) + A(x)^2*(x^4 + x^5 + x^6 + x^7 + x^8) + A(x)^3*(x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15) +...

Prog

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

Crossrefs

Cf. A199410.

Sequence in context: A136671 A274114 A024557 * A025241 A292461 A203019

Adjacent sequences: A199406 A199407 A199408 * A199410 A199411 A199412

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2011

Status

approved