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A318644 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n*(n+1)/2) * x^n / A(x)^n.

%I A318644 #13 Apr 03 2019 00:41:17
%S A318644 1,1,1,1,2,4,11,32,106,376,1433,5782,24574,109393,508026,2453256,
%T A318644 12285347,63656731,340626704,1879183856,10672897341,62323897482,
%U A318644 373748877678,2299318074357,14497472040378,93599428822052,618278575554155,4175348680420942,28806364292660618,202899326988089615,1458130019936912105,10685096640964659318
%N A318644 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n*(n+1)/2) * x^n / A(x)^n.
%H A318644 Paul D. Hanna, <a href="/A318644/b318644.txt">Table of n, a(n) for n = 0..495</a>
%F A318644 G.f. A(x) satisfies:
%F A318644 (1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / A(x)^n.
%F A318644 (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / A(x)^n.
%e A318644 G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + 24574*x^12 + 109393*x^13 + 508026*x^14 + ...
%e A318644 such that
%e A318644 A(x) = 1 + (1+x)*x/A(x) + (1+x)^3*x^2/A(x)^2 + (1+x)^6*x^3/A(x)^3 + (1+x)^10*x^4/A(x)^4 + (1+x)^15*x^5/A(x)^5 + (1+x)^21*x^6/A(x)^6 + (1+x)^28*x^7/A(x)^7 + ... + (1+x)^(n*(n+1)/2) * x^n / A(x)^n + ...
%e A318644 Also
%e A318644 1 + x = 1 + x/A(x) + (1+x)*x^2/A(x)^2 + (1+x)^3*x^3/A(x)^3 + (1+x)^6*x^4/A(x)^4 + (1+x)^10*x^5/A(x)^5 + (1+x)^15*x^6/A(x)^6 + (1+x)^21*x^7/A(x)^7 + ...
%o A318644 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, (1+x +x*O(x^#A))^(n*(n+1)/2) * x^n/Ser(A)^n ) )[#A] ); A[n+1]}
%o A318644 for(n=0, 30, print1(a(n), ", "))
%Y A318644 Cf. A320951, A303058.
%K A318644 nonn
%O A318644 0,5
%A A318644 _Paul D. Hanna_, Sep 07 2018