A204858 G.f.: Sum_{n>=0} n! * x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).

1, 1, 1, 3, 3, 7, 13, 21, 33, 67, 115, 183, 333, 541, 937, 1635, 2643, 4327, 7573, 12069, 20025, 33427, 54259, 87375, 144669, 231541, 374809, 607443, 970539, 1545367, 2502205, 3947541, 6270057, 9997867, 15776083, 24832503, 39351309, 61552501, 96632689

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..6263

Formula

G.f.: 1/(1 - x/(1 - 2*x^2*(1-x)/(1 - 3*x^3*(1-2*x^2)/(1 - 4*x^4*(1-3*x^3)/(1 - 5*x^5*(1-4*x^4)/(1 - 6*x^6*(1-5*x^5)/(1 -...)))))), a continued fraction.

From Vaclav Kotesovec, Jun 18 2019: (Start)

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

c = 8007.60951343849770902289074154120578227939552369... if mod(n,3)=0

c = 8007.30566699919825273673656299755925992856381905... if mod(n,3)=1

c = 8007.19663204881021378993302255541874790731157021... if mod(n,3)=2

(End)

Example

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 21*x^7 +...
where A(x) = 1 + x/(1-x) + 2!*x^3/((1-x)*(1-2*x^2)) + 3!*x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + 4!*x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...

Mathematica

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

Prog

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

Crossrefs

Cf. A032020, A204857, A204856.

Sequence in context: A187873 A306665 A065876 * A095008 A214825 A230017

Adjacent sequences: A204855 A204856 A204857 * A204859 A204860 A204861

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2012

Status

approved