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A326558 G.f. A(x) satisifes: Sum_{n>=0} A(x)^((n-1)^2) * x^n = Sum_{n>=0} ((1+x)^(n-1) + 1)^n * x^n. 0

%I #5 Sep 21 2019 14:35:09

%S 1,1,3,10,33,126,535,2493,12569,68092,394029,2423257,15767827,

%T 108113333,778198815,5859990774,46016573248,375739918599,

%U 3181928792057,27881971943496,252291917965337,2353166240022595,22588471117451579,222844380918633385,2256632985093301978,23430724178641674376,249198423317854694260,2712372210346517891943

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

%F G.f. A(x) satisifes:

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

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

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 33*x^4 + 126*x^5 + 535*x^6 + 2493*x^7 + 12569*x^8 + 68092*x^9 + 394029*x^10 + 2423257*x^11 + 15767827*x^12 + ...

%e such that the following series are all 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)*x + (1 + (1+x))^2*x^2 + (1 + (1+x)^2)^3*x^3 + (1 + (1+x)^3)^4*x^4 + (1 + (1+x)^4)^5*x^5 + ... + (1 + (1+x)^(n-1))^n*x^n + ...

%e also

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

%e where

%e B(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 41*x^4 + 164*x^5 + 728*x^6 + 3546*x^7 + 18679*x^8 + 105445*x^9 + 633198*x^10 + 4021124*x^11 + 26876020*x^12 + ...

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

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

%Y Cf. A326557.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 16 2019

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