A219331 L.g.f.: -log(1 - Sum_{n>=1} x^(n^2)) = Sum_{n>=1} a(n)*x^n/n.

1, 1, 1, 5, 6, 7, 8, 13, 28, 36, 45, 59, 92, 134, 186, 269, 375, 538, 761, 1080, 1520, 2157, 3060, 4339, 6181, 8750, 12394, 17554, 24912, 35322, 50066, 70957, 100596, 142665, 202278, 286790, 406520, 576347, 817142, 1158528, 1642461, 2328536, 3301283, 4680417, 6635688

Offset: 1

Author

Paul D. Hanna, Apr 12 2013

Keywords

• nonn

Comments

Limit a(n)/a(n+1) = 0.705346681379806989636379706393941505260078161512292870... is a real root of 1 = Sum_{n>=1} x^(n^2).

Examples

			L.g.f.: L(x) = x + x^2/2 + x^3/3 + 5*x^4/4 + 6*x^5/5 + 7*x^6/6 + 8*x^7/7 + 13*x^8/8 + 28*x^9/9 + 36*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 11*x^9 + 16*x^10 + 22*x^11 + 30*x^12 +...+ A006456(n)*x^n +...
exp(-L(x)) = 1 - x - x^4 - x^9 - x^16 - x^25 - x^36 +...+ -x^(n^2) +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A224607, A224608, A006456.

Programs

PARI

{a(n)=n*polcoeff(-log(1-sum(r=1,sqrtint(n+1),x^(r^2)+x*O(x^n))),n)}
for(n=1,50,print1(a(n),", "))

Formula

Logarithmic derivative of A006456, where A006456(n) is the number of compositions of n into sums of squares.