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