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%I #12 Jun 02 2019 19:30:32
%S 1,1,2,1,1,3,1,1,5,1,1,5,1,4,6,1,1,7,8,1,8,1,1,19,1,5,10,1,16,11,1,1,
%T 23,22,1,13,1,1,42,21,1,20,1,37,16,1,36,17,46,1,34,1,1,130,1,1,20,1,
%U 67,56,85,7,22,79,1,23,1,121,185,1,1,25,23,106,191,1,1
%N G.f.: Sum_{n>=0} (x + x^(2*n))^n.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,
%C (2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),
%C for any fixed integer k; here, k = 1 and y = 1, F = x, G = x^2.
%H Paul D. Hanna, <a href="/A217667/b217667.txt">Table of n, a(n) for n = 0..1000</a>
%F Generating functions.
%F (1) Sum_{n>=0} (x + x^(2*n))^n.
%F (2) Sum_{n>=0} x^(n^2) / (1 - x^(2*n+1))^(n+1). - _Paul D. Hanna_, Jun 02 2019
%e G.f.: A(x) = 1 + x + 2*x^2 + x^3 + x^4 + 3*x^5 + x^6 + x^7 + 5*x^8 + x^9 +...
%e where
%e A(x) = 1 + (x + x^2) + (x + x^4)^2 + (x + x^6)^3 + (x + x^8)^4 + (x + x^10)^5 +...
%e Also
%e A(x) = 1/(1-x) + x/(1 - x^3)^2 + x^4/(1 - x^5)^3 + x^9/(1 - x^7)^4 + x^16/(1 - x^9)^5 + x^25/(1 - x^11)^6 + x^36/(1 - x^13)^7 + x^49/(1 - x^15)^8 + ...
%t terms = 100; Sum[(x + x^(2*n))^n, {n, 0, terms}] + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, May 16 2017 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,(x+x^(2*m) +x*O(x^n))^m),n)}
%o for(n=0,100,print1(a(n),", "))
%Y Cf. A217668, A217669.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 10 2012
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