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%I #3 Jun 01 2019 06:20:08
%S 1,0,1,0,1,1,3,10,31,121,464,1944,8454,38468,182126,893488,4535670,
%T 23760888,128267430,712403572,4065752904,23816376636,143051516760,
%U 880239634009,5544258942957,35718401802001,235202635677715,1582012735794119,10862478047272181,76093536057355965,543536686935606339,3956823673660817241,29341805120002375853,221536339165494454489,1702261439852726415968,13305909830342110613840,105760138628395361333444
%N G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / A(x)^(n*(n+1)/2).
%C Compare to: 1+x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / G(x)^(n*(n+1)/2) holds when G(x) = (1+x).
%e G.f.: A(x) = 1 + x^2 + x^4 + x^5 + 3*x^6 + 10*x^7 + 31*x^8 + 121*x^9 + 464*x^10 + 1944*x^11 + 8454*x^12 + 38468*x^13 + 182126*x^14 + 893488*x^15 + ...
%e such that
%e 1/(1-x) = 1 + x/A(x) + x^2*(1+x)/A(x)^3 + x^3*(1+x)^3/A(x)^6 + x^4*(1+x)^6/A(x)^10 + x^5*(1+x)^10/A(x)^15 + x^6*(1+x)^15/A(x)^21 + x^7*(1+x)^21/A(x)^28 + x^8*(1+x)^28/A(x)^36 + x^9*(1+x)^36/A(x)^45 + ...
%o (PARI) a(n)=my(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=0,#A,x^m*((1+x+x*O(x^#A))^(m*(m-1)/2)/Ser(A)^(m*(m+1)/2)-1)),#A));A[n+1]
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A325578.
%K nonn
%O 0,7
%A _Paul D. Hanna_, Jun 01 2019
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