A206739 G.f.: 1/(1 - x/(1 - x^4/(1 - x^9/(1 - x^16/(1 - x^25/(1 - x^36/(1 -...- x^(n^2)/(1 -...))))))), a continued fraction.

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 37, 52, 72, 99, 138, 193, 269, 373, 518, 722, 1006, 1399, 1944, 2705, 3766, 5241, 7290, 10141, 14112, 19638, 27323, 38012, 52889, 73593, 102398, 142470, 198225, 275809, 383760, 533954, 742923, 1033685, 1438254

Offset

0,6

Links

Table of n, a(n) for n=0..46.

Formula

a(n) = sum(k=0..n, T(n,k)), where T(n, m)=sum(i=1..(n-m)/3, binomial(m, i)*T((n-m)/3,i)), T(n,n)=1. - Vladimir Kruchinin, Mar 21 2015

G.f.: A(x)=1/B(x), where B(x) is g.f. of A290975. - Seiichi Manyama, Aug 18 2017

a(n) ~ c * d^n, where d = 1.391377080590304271048017099353... and c = 0.3625537262803710555422183139... - Vaclav Kotesovec, Aug 24 2017

Example

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

Mathematica

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

Prog

(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)^2)*CF)); polcoeff(CF, n, x)}
for(n=0, 55, print1(a(n), ", "))
(PARI) N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^((k+1)^2) / G(k+1) );
gf = 1 / G(0);
Vec(gf) \\ Joerg Arndt, Jul 06 2013
(Maxima)
T(n, m):=if n=m then 1 else  sum(binomial(m, i)*T((n-m)/3, i), i, 1, (n-m)/3);
makelist(sum(T(n, k), k, 0, n), n, 0, 20); /* Vladimir Kruchinin, Mar 21 2015 */

Crossrefs

Cf. A206740, A290975.

Sequence in context: A087221 A352041 A295072 * A107586 A206737 A275174

Adjacent sequences: A206736 A206737 A206738 * A206740 A206741 A206742

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2012

Status

approved