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G.f. A(x) satisfies: Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} (A(x)^(n-2) + 1)^n * x^n.
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%I #7 Aug 24 2019 23:01:19

%S 1,1,2,5,19,87,458,2650,16459,108313,748530,5400001,40494822,

%T 314699916,2528714512,20972984213,179305049748,1578420640715,

%U 14293912578588,133053118062054,1272096858349455,12483241986776063,125644852173512082,1296202208475495630,13696609133818915790,148138542880240522733,1638862202896363931279,18532964342704899428439,214086944190169455544290,2524648814791758862995327

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

%H Paul D. Hanna, <a href="/A326563/b326563.txt">Table of n, a(n) for n = 0..250</a>

%F G.f. A(x) allows the following sums to be equal:

%F (1) B(x) = Sum_{n>=0} A(x)^((n-1)^2) * x^n.

%F (2) B(x) = Sum_{n>=0} (A(x)^(n-2) + 1)^n * x^n.

%F (3) B(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / (1 - x*A(x)^n)^(n+1).

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 87*x^5 + 458*x^6 + 2650*x^7 + 16459*x^8 + 108313*x^9 + 748530*x^10 + 5400001*x^11 + 40494822*x^12 + ...

%e such that the following sums are equal

%e B(x) = A(x) + x + A(x)*x^2 + A(x)^4*x^3 + A(x)^9*x^4 + A(x)^16*x^5 + A(x)^25*x^6 + A(x)^36*x^7 + A(x)^49*x^8 + ... + A(x)^((n-1)^2)*x^n + ...

%e and

%e B(x) = 1 + (1 + 1/A(x))*x + 2^2*x^2 + (1 + A(x))^3*x^3 + (1 + A(x)^2)^4*x^4 + (1 + A(x)^3)^5*x^5 + (1 + A(x)^4)^6*x^6 + ... + (1 + A(x)^(n-2))^n*x^n + ...

%e also

%e B(x) = 1/(1 - x) + 1/A(x)*x/(1 - x*A(x))^2 + x^2/(1 - x*A(x)^2)^3 + A(x)^3*x^3/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n-2))*x^n/(1 - x*A(x)^n)^(n+1) + ...

%e where

%e B(x) = 1 + 2*x + 3*x^2 + 7*x^3 + 26*x^4 + 116*x^5 + 596*x^6 + 3373*x^7 + 20541*x^8 + 132803*x^9 + 903151*x^10 + 6420523*x^11 + 47502514*x^12 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m-2) + 1)^m*x^m - Ser(A)^((m-1)^2)*x^m ), #A-1)); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A326560, A326561, A326562, A326275, A326287.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 24 2019