A224678 L.g.f.: -log(1 - Sum_{n>=1} x^(n*(n+1)/2)) = Sum_{n>=1} a(n)*x^n/n.

1, 1, 4, 5, 6, 16, 22, 29, 49, 86, 122, 188, 300, 456, 714, 1117, 1718, 2653, 4124, 6390, 9916, 15368, 23806, 36884, 57181, 88622, 137344, 212896, 329934, 511316, 792516, 1228285, 1903598, 2950334, 4572602, 7086833, 10983562, 17022956, 26382984, 40889694, 63373086, 98218920

Offset: 1

Author

Paul D. Hanna, Apr 14 2013

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 + 5*x^4/4 + 6*x^5/5 + 16*x^6/6 + 22*x^7/7 + 29*x^8/8 + 49*x^9/9 + 86*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 11*x^7 + 16*x^8 + 25*x^9 + 40*x^10 + 61*x^11 + 94*x^12 +...+ A023361(n)*x^n +...
exp(-L(x)) = 1 - x - x^3 - x^6 - x^10 - x^21 - x^28 +...+ -x^(n*(n+1)/2) +...
		

Crossrefs

Cf. A023361, A219331, A224680.

Programs

PARI

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

Formula

Logarithmic derivative of A023361, where A023361(n) = number of compositions of n into positive triangular numbers.