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A326088
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G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2) * x^n = Sum_{n>=0} ((1+x)^n - 1)^n.
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0
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1, 3, 18, 184, 2742, 51650, 1148054, 29089167, 823981958, 25773170170, 882457387327, 32850733172032, 1322072236117388, 57235976215014221, 2653750194041974871, 131246449114771366495, 6898798056206294405352, 384135589510998920667366, 22590113301696105008398833, 1399232160794278896505982537, 91058022280629233139175411272
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OFFSET
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0,2
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LINKS
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FORMULA
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The g.f. A(x) allows the following series to be equal:
(1) B(x) = Sum_{n>=0} A(x)^(n^2) * x^n.
(2) B(x) = Sum_{n>=0} ((1+x)^n - 1)^n.
(3) B(x) = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n)^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 18*x^2 + 184*x^3 + 2742*x^4 + 51650*x^5 + 1148054*x^6 + 29089167*x^7 + 823981958*x^8 + 25773170170*x^9 + 882457387327*x^10 + ...
such that the following sum
B(x) = 1 + A(x)*x + A(x)^4*x^2 + A(x)^9*x^3 + A(x)^16*x^4 + A(x)^25*x^5 + A(x)^36*x^6 + A(x)^49*x^7 + A(x)^64*x^8 + ... + A(x)^(n^2)*x^n + ...
equals
B(x) = 1 + ((1+x) - 1) + ((1+x)^2 - 1)^2 + ((1+x)^3 - 1)^3 + ((1+x)^4 - 1)^4 + ((1+x)^5 - 1)^5 + ((1+x)^6 - 1)^6 + ... + ((1+x)^n - 1)^n + ...
as well as
B(x) = 1/2 + (1+x)/(1 + (1+x))^2 + (1+x)^4/(1 + (1+x)^2)^3 + (1+x)^9/(1 + (1+x)^3)^4 + (1+x)^16/(1 + (1+x)^4)^5 + ... + (1+x)^(n^2)/(1 + (1+x)^n)^(n+1) + ...
where
B(x) = 1 + x + 4*x^2 + 31*x^3 + 338*x^4 + 4769*x^5 + 82467*x^6 + 1687989*x^7 + 39905269*x^8 + 1069863695*x^9 + ... + A122400(n)*x^n + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, ((1+x)^m - 1 +x*O(x^#A))^m - x^m*Ser(A)^(m^2) ), #A)); 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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