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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+2))^n * x^n.
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%I #17 Jul 24 2019 20:05:22

%S 1,1,2,9,53,357,2623,20487,167571,1421663,12430232,111506176,

%T 1022965373,9575262402,91290592979,885415696549,8728300482538,

%U 87399936654843,888643475396035,9172662449948948,96116312656950277,1022513226629210310,11045623068915556941,121191713354394285810,1350988071125342019102,15306389786282149439384,176313553377393691773895

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

%H Paul D. Hanna, <a href="/A326287/b326287.txt">Table of n, a(n) for n = 0..300</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 + 9*x^3 + 53*x^4 + 357*x^5 + 2623*x^6 + 20487*x^7 + 167571*x^8 + 1421663*x^9 + 12430232*x^10 + ...

%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)^64*x^7 + ...

%e and

%e B(x) = 1 + (1 + A(x)^3)*x + (1 + A(x)^4)^2*x^2 + (1 + A(x)^5)^3*x^3 + (1 + A(x)^6)^4*x^4 + (1 + A(x)^7)^5*x^5 + (1 + A(x)^8)^6*x^6 + (1 + A(x)^9)^7*x^7 + ...

%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)^20*x/(1 - x*A(x)^4)^5 + ...

%e where

%e B(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 188*x^4 + 1213*x^5 + 8537*x^6 + 64061*x^7 + 505060*x^8 + 4142471*x^9 + 35103445*x^10 + 305821769*x^11 + 2729269060*x^12 + 24884029770*x^13 + 231326144627*x^14 + 2189345765437*x^15 + 21072597189266*x^16 + 206111127679609*x^17 + 2047583929329250*x^18 + ...

%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. A326275, A326560, A326561, A326562, A326563.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 12 2019