A206739 - OEIS

%I #24 Aug 24 2017 16:52:37

%S 1,1,1,1,1,2,3,4,5,7,10,14,19,26,37,52,72,99,138,193,269,373,518,722,

%T 1006,1399,1944,2705,3766,5241,7290,10141,14112,19638,27323,38012,

%U 52889,73593,102398,142470,198225,275809,383760,533954,742923,1033685,1438254

%N 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.

%F 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

%F G.f.: A(x)=1/B(x), where B(x) is g.f. of A290975.  - _Seiichi Manyama_, Aug 18 2017

%F a(n) ~ c * d^n, where d = 1.391377080590304271048017099353... and c = 0.3625537262803710555422183139... - _Vaclav Kotesovec_, Aug 24 2017

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

%t 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 *)

%o (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)}

%o for(n=0,55,print1(a(n),", "))

%o (PARI) N = 66;  q = 'q + O('q^N);

%o G(k) = if(k>N, 1, 1 - q^((k+1)^2) / G(k+1) );

%o gf = 1 / G(0);

%o Vec(gf) \\ _Joerg Arndt_, Jul 06 2013

%o (Maxima)

%o T(n, m):=if n=m then 1 else  sum(binomial(m, i)*T((n-m)/3, i), i, 1, (n-m)/3);

%o makelist(sum(T(n,k),k,0,n),n,0,20); /* _Vladimir Kruchinin_, Mar 21 2015 */

%Y Cf. A206740, A290975.

%K nonn

%O 0,6

%A _Paul D. Hanna_, Feb 12 2012