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G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n-1)+1) * x^n = Sum_{n>=0} (A(x)^(n-1) + 1)^n * x^n.
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%I #9 Jul 24 2019 19:59:10

%S 1,1,2,6,26,134,775,4856,32359,226688,1656745,12566075,98550684,

%T 797062686,6635504831,56781439908,498946126131,4498769217830,

%U 41598980113524,394300130478239,3829670219184141,38100703672385734,388140101200555331,4047307115413115559,43180809390468971803,471163737793390252840,5255377025199543952036,59891933515705554763680,697006510462415153074548

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

%H Paul D. Hanna, <a href="/A326562/b326562.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*(n-1)+1) * 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 + 2*x^2 + 6*x^3 + 26*x^4 + 134*x^5 + 775*x^6 + 4856*x^7 + 32359*x^8 + 226688*x^9 + 1656745*x^10 + 12566075*x^11 + 98550684*x^12 + ...

%e such that the following sums are equal

%e B(x) = A(x) + A(x)*x + A(x)^3*x^2 + A(x)^7*x^3 + A(x)^13*x^4 + A(x)^21*x^5 + A(x)^31*x^6 + A(x)^43*x^7 + A(x)^57*x^8 + ... + A(x)^(n*(n-1)+1)*x^n + ...

%e and

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

%e also

%e B(x) = 1/(1 - x) + x/(1 - x*A(x))^2 + A(x)^2*x^2/(1 - x*A(x)^2)^3 + A(x)^6*x^3/(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 + 4*x^2 + 12*x^3 + 49*x^4 + 240*x^5 + 1328*x^6 + 8014*x^7 + 51691*x^8 + 351839*x^9 + 2505762*x^10 + 18563322*x^11 + 142460948*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^2-m+1)*x^m ), #A-1)); A[n+1]}

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

%Y Cf. A326560, A326561, A326563, A326275, A326287.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 23 2019