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A320951 G.f.: A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.

%I A320951 #11 Nov 29 2018 12:16:14
%S A320951 1,1,2,3,9,28,110,485,2358,12486,70726,425747,2702837,18004835,
%T A320951 125337381,908737863,6843536374,53407750147,431075414218,
%U A320951 3592384229312,30862831600689,272976843937138,2482698463801148,23192576636266041,222310388884578760,2184486850658804107,21985733344615744620,226455749821063728474,2385331864619907236147,25676170688883138634306,282253492062060457638824
%N A320951 G.f.: A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.
%H A320951 Paul D. Hanna, <a href="/A320951/b320951.txt">Table of n, a(n) for n = 0..200</a>
%F A320951 G.f. A(x) satisfies:
%F A320951 (1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)) / A(x)^n.
%F A320951 (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)) / A(x)^n.
%F A320951 (3) A(x) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n-1)) / A(A(x)-1)^n.
%e A320951 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 28*x^5 + 110*x^6 + 485*x^7 + 2358*x^8 + 12486*x^9 + 70726*x^10 + 425747*x^11 + 2702837*x^12 + ...
%e A320951 such that
%e A320951 A(x) = 1 + x*(1+x)^2/A(x) + x^2*(1+x)^6/A(x)^2 + x^3*(1+x)^12/A(x)^3 + x^4*(1+x)^20/A(x)^4 + x^5*(1+x)^30/A(x)^5 + ...
%e A320951 Also
%e A320951 1 + x = 1 + x/A(x) + x^2*(1+x)^2/A(x)^2 + x^3*(1+x)^6/A(x)^3 + x^4*(1+x)^12/A(x)^4 + x^5*(1+x)^20/A(x)^5 + x^6*(1+x)^30/A(x)^6 + ...
%e A320951 RELATED SERIES.
%e A320951 Sum_{n>=0} x^n * (1+x)^(n^2) / A(x)^n = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 32*x^6 + 135*x^7 + 649*x^8 + 3381*x^9 + 18894*x^10 + 112382*x^11 + 705174*x^12 + ...
%e A320951 A(A(x)-1) = 1 + x + 4*x^2 + 14*x^3 + 56*x^4 + 251*x^5 + 1239*x^6 + 6627*x^7 + 38112*x^8 + 233692*x^9 + 1517788*x^10 + 10384824*x^11 + ...
%e A320951 where A(A(x)-1) = Sum_{n>=0} (A(x)-1)^n * A(x)^(n*(n+1)) / A(A(x)-1)^n.
%o A320951 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n +x*O(x^#A))^(n+1) * x^n/Ser(A)^n ) )[#A] ); A[n+1]}
%o A320951 for(n=0, 30, print1(a(n), ", "))
%Y A320951 Cf. A318644, A303058.
%K A320951 nonn
%O A320951 0,3
%A A320951 _Paul D. Hanna_, Nov 20 2018