A206740 G.f.: 1/(1 - x/(1 - x^3/(1 - x^6/(1 - x^10/(1 - x^15/(1 - x^21/(1 -...- x^(n*(n+1)/2)/(1 -...))))))), a continued fraction.

1, 1, 1, 1, 2, 3, 4, 6, 9, 13, 20, 30, 44, 66, 99, 147, 220, 329, 490, 732, 1095, 1634, 2440, 3646, 5444, 8130, 12146, 18139, 27089, 40463, 60434, 90258, 134811, 201349, 300721, 449153, 670844, 1001939, 1496467, 2235080, 3338227, 4985868, 7446739, 11122179

Offset: 0

Author

Paul D. Hanna, Feb 12 2012

Keywords

• nonn

Examples

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

Crossrefs

Cf. A206739.

Programs

Mathematica

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1]*(Range[nmax + 1] + 1)/2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2017 *)

PARI

{a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^((M-k+1)*(M-k+2)/2)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

Formula

G.f.: 1/Q(0) , where Q(k) = 1 - x^((2*k+1)*(2*k+2)/2)/(1 - x^((2*k+2)*(2*k+3)/2)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 10 2013

a(n) ~ c * d^n, where d = 1.49356638691558702616975760297981328... and c = 0.35853801643147450974166770910994348... - Vaclav Kotesovec, Aug 24 2017