A204856 G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).

1, 1, 1, 2, 2, 4, 5, 9, 11, 22, 27, 49, 68, 115, 157, 279, 372, 628, 914, 1457, 2070, 3457, 4840, 7753, 11442, 17768, 25824, 41315, 59008, 92140, 137212, 208524, 305472, 477659, 691381, 1058019, 1575694, 2370618, 3491693, 5359888, 7796346, 11799263, 17583757

Offset: 0

Author

Paul D. Hanna, Jan 20 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 9*x^7 + 11*x^8 +...
where A(x) = 1 + x/(1-x) + x^3/((1-x)*(1-2*x^2)) + x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..6279 (terms 0..250 from Vaclav Kotesovec)

Crossrefs

Cf. A204855, A204858.

Column sums of A367955.

Programs

Mathematica

Table[SeriesCoefficient[Sum[x^Binomial[n + 1, 2]/Product[(1 - k*x^k), {k, 1, n}], {x, 0, 100}], {x, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 19 2017 *)

PARI

{a(n)=polcoeff(1+sum(m=1,n,x^(m*(m+1)/2)/prod(k=1,m,1-k*x^k+x*O(x^n))),n)}

Formula

G.f.: 1/(1 - x/(1 - x^2*(1-x)/(1-x^2 - x^3*(1-2*x^2)/(1-2*x^3 - x^4*(1-3*x^3)/(1-3*x^4 - x^5*(1-4*x^4)/(1-4*x^5 -...)))))), a continued fraction.

From Vaclav Kotesovec, Jun 18 2019: (Start)

a(n) ~ c * 3^(n/3), where

c = 23.5612420584121380174441491950859168338330954540437... if mod(n,3)=0

c = 23.5209031427848763179214171003561794127717213180726... if mod(n,3)=1

c = 23.5214569018665529984420312927586688667133017590049... if mod(n,3)=2

(End)