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A321602
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G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(2*n) - A(x))^n.
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6
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1, 2, 5, 68, 1521, 45328, 1660032, 71548008, 3533826841, 196432984748, 12128132342482, 823366216285428, 60966207548525287, 4890600994792550264, 422601696583826709492, 39142599000082019249968, 3869325702147169825040193, 406650337650126697706078146, 45281361448272561712508294157, 5325916931170845646048163850556, 659842223101960470758187538118437
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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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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(2*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(2*n^2) / (1 + (1+x)^(2*n)*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 5*x^2 + 68*x^3 + 1521*x^4 + 45328*x^5 + 1660032*x^6 + 71548008*x^7 + 3533826841*x^8 + 196432984748*x^9 + 12128132342482*x^10 + ...
such that
1 = 1 + ((1+x)^2 - A(x)) + ((1+x)^4 - A(x))^2 + ((1+x)^6 - A(x))^3 + ((1+x)^8 - A(x))^4 + ((1+x)^10 - A(x))^5 + ((1+x)^12 - A(x))^6 + ((1+x)^14 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)^2/(1 + (1+x)^2*A(x))^2 + (1+x)^8/(1 + (1+x)^4*A(x))^3 + (1+x)^18/(1 + (1+x)^6*A(x))^4 + (1+x)^32/(1 + (1+x)^8*A(x))^5 + (1+x)^50/(1 + (1+x)^10*A(x))^6 + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 2 + 6*x + 182*x^2 + 5554*x^3 + 211172*x^4 + 9397920*x^5 + 476737830*x^6 + 27086036234*x^7 + 1702330030676*x^8 + ...
the coefficients of which are all even:
(1/2) * A'(x)/A(x) = 1 + 3*x + 91*x^2 + 2777*x^3 + 105586*x^4 + 4698960*x^5 + 238368915*x^6 + 13543018117*x^7 + 851165015338*x^8 + ...
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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] = Vec( sum(m=0, #A, ((1+x)^(2*m) - Ser(A))^m ) )[#A] ); H=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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