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A326557 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. 2

%I #9 Sep 21 2019 14:34:34

%S 1,1,1,1,3,12,64,391,2617,18738,141483,1116801,9160502,77745060,

%T 680550918,6129635386,56699324213,537823602765,5225075478099,

%U 51939709551433,527829047648887,5479728265490353,58079392804968241,628114208288086710,6927692801388774583,77887967322146451681,892270205641708989800,10410949755661589229619

%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.

%H Paul D. Hanna, <a href="/A326557/b326557.txt">Table of n, a(n) for n = 0..200</a>

%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 + x^2 + x^3 + 3*x^4 + 12*x^5 + 64*x^6 + 391*x^7 + 2617*x^8 + 18738*x^9 + 141483*x^10 + 1116801*x^11 + 9160502*x^12 + ...

%e such that the following series are all 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 + ... + A(x)^((n+1)^2) * x^n + ...

%e and

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

%e also

%e B(x) = 1/(1 - x) + (1+x)^2*x/(1 - x*(1+x))^2 + (1+x)^6*x^2/(1 - x*(1+x)^2)^3 + (1+x)^12*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 + 6*x^2 + 21*x^3 + 85*x^4 + 382*x^5 + 1879*x^6 + 9986*x^7 + 56818*x^8 + 343640*x^9 + 2196596*x^10 + ... + A326276(n)*x^n + ...

%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. A326276.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Sep 14 2019

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Last modified July 18 06:54 EDT 2024. Contains 374377 sequences. (Running on oeis4.)