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A326557
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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.
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2
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1, 1, 1, 1, 3, 12, 64, 391, 2617, 18738, 141483, 1116801, 9160502, 77745060, 680550918, 6129635386, 56699324213, 537823602765, 5225075478099, 51939709551433, 527829047648887, 5479728265490353, 58079392804968241, 628114208288086710, 6927692801388774583, 77887967322146451681, 892270205641708989800, 10410949755661589229619
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f. A(x) satisifes:
(1) Sum_{n>=0} A(x)^((n+1)^2) * x^n = Sum_{n>=0} ((1+x)^(n+1) + 1)^n * x^n.
(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).
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EXAMPLE
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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 + ...
such that the following series are all equal
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 + ...
and
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 + ...
also
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) + ...
where
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 + ...
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PROG
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(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]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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