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G.f. A(x) satisfies: Sum_{n>=0} A(x)^((n+1)^2) * x^n = Sum_{n>=0} (1 + A(x)^(n+1))^n * x^n.
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%I #17 Jul 23 2019 22:10:43

%S 1,1,1,3,15,85,535,3623,25951,194520,1514905,12198161,101200455,

%T 862904340,7548194457,67646260677,620496791884,5821278151196,

%U 55827806529613,547097673373322,5476831304418641,55993353247707012,584508694490025552,6228747205436059856,67743826781449323262,751784959486903813202

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

%H Paul D. Hanna, <a href="/A326275/b326275.txt">Table of n, a(n) for n = 0..200</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+1) + 1)^n * x^n.

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

%e G.f. A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 85*x^5 + 535*x^6 + 3623*x^7 + 25951*x^8 + 194520*x^9 + 1514905*x^10 + 12198161*x^11 + 101200455*x^12 + ...

%e such that the following sums are equal

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

%e and

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

%e also

%e B(x) = 1/(1 - x) + A(x)^2*x/(1 - x*A(x))^2 + A(x)^6*x/(1 - x*A(x)^2)^3 + A(x)^12*x/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n+1))*x^n/(1 - x*A(x)^n)^(n+1) + ...

%e where

%e B(x) = 1 + 2*x + 6*x^2 + 23*x^3 + 105*x^4 + 545*x^5 + 3118*x^6 + 19261*x^7 + 126615*x^8 + 876553*x^9 + 6342647*x^10 + 47701975*x^11 + 371337731*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+1) + 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. A326287, A326560, A326561, A326562.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jun 28 2019