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A326088 G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2) * x^n = Sum_{n>=0} ((1+x)^n - 1)^n.

%I A326088 #3 Jun 28 2019 17:11:53
%S A326088 1,3,18,184,2742,51650,1148054,29089167,823981958,25773170170,
%T A326088 882457387327,32850733172032,1322072236117388,57235976215014221,
%U A326088 2653750194041974871,131246449114771366495,6898798056206294405352,384135589510998920667366,22590113301696105008398833,1399232160794278896505982537,91058022280629233139175411272
%N A326088 G.f. A(x) satisfies:  Sum_{n>=0} A(x)^(n^2) * x^n  =  Sum_{n>=0} ((1+x)^n - 1)^n.
%F A326088 The g.f. A(x) allows the following series to be equal:
%F A326088 (1) B(x) = Sum_{n>=0} A(x)^(n^2) * x^n.
%F A326088 (2) B(x) = Sum_{n>=0} ((1+x)^n - 1)^n.
%F A326088 (3) B(x) = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1).
%e A326088 G.f.: A(x) = 1 + 3*x + 18*x^2 + 184*x^3 + 2742*x^4 + 51650*x^5 + 1148054*x^6 + 29089167*x^7 + 823981958*x^8 + 25773170170*x^9 + 882457387327*x^10 + ...
%e A326088 such that the following sum
%e A326088 B(x) = 1 + A(x)*x + A(x)^4*x^2 + A(x)^9*x^3 + A(x)^16*x^4 + A(x)^25*x^5 + A(x)^36*x^6 + A(x)^49*x^7 + A(x)^64*x^8 + ... + A(x)^(n^2)*x^n + ...
%e A326088 equals
%e A326088 B(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)^2 + ((1+x)^3 - 1)^3 + ((1+x)^4 - 1)^4 + ((1+x)^5 - 1)^5 + ((1+x)^6 - 1)^6 + ... + ((1+x)^n - 1)^n + ...
%e A326088 as well as
%e A326088 B(x) = 1/2 + (1+x)/(1 + (1+x))^2 + (1+x)^4/(1 + (1+x)^2)^3 + (1+x)^9/(1 + (1+x)^3)^4 + (1+x)^16/(1 + (1+x)^4)^5 + ... + (1+x)^(n^2)/(1 + (1+x)^n)^(n+1) + ...
%e A326088 where
%e A326088 B(x) = 1 + x + 4*x^2 + 31*x^3 + 338*x^4 + 4769*x^5 + 82467*x^6 + 1687989*x^7 + 39905269*x^8 + 1069863695*x^9 + ... + A122400(n)*x^n + ...
%o A326088 (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0);
%o A326088 A[#A] = polcoeff( sum(m=0,#A, ((1+x)^m - 1  +x*O(x^#A))^m - x^m*Ser(A)^(m^2) ), #A)); A[n+1]}
%o A326088 for(n=0,30, print1(a(n),", "))
%Y A326088 Cf. A122400.
%K A326088 nonn
%O A326088 0,2
%A A326088 _Paul D. Hanna_, Jun 28 2019