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A317997
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G.f. A(x) satisfies: Sum_{n>=0} ( x^n + (-1)^n*A(x) )^n = 1.
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9
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1, 1, 3, 8, 22, 62, 184, 566, 1779, 5678, 18367, 60091, 198454, 660659, 2214700, 7469730, 25330135, 86308528, 295349232, 1014609122, 3497707471, 12096300263, 41955256710, 145908382056, 508676792788, 1777418947825, 6223737747078, 21835417474588, 76747151840311, 270209860057674, 952864014673991, 3365181832760454, 11901315653535352
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( x^n + (-1)^n*A(x) )^n.
(2) 1 = Sum_{n>=0} x^(n^2) / (1 + (-x)^n*A(x))^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 3.699932925974338637547631722109256130661... and c = 0.41429291356124695745726003609883876... - Vaclav Kotesovec, Sep 06 2018
A(1/d) = 0.8093560777550741... where d is given above. - Paul D. Hanna, Oct 08 2019
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 22*x^5 + 62*x^6 + 184*x^7 + 566*x^8 + 1779*x^9 + 5678*x^10 + 18367*x^11 + 60091*x^12 + ...
such that
1 = 1 + (x - A(x)) + (x^2 + A(x))^2 + (x^3 - A(x))^3 + (x^4 + A(x))^4 + (x^5 - A(x))^5 + (x^6 + A(x))^6 + (x^7 - A(x))^7 + ...
Also
1 = 1/(1 + A(x)) + x/(1 - x*A(x))^2 + x^4/(1 + x^2*A(x))^3 + x^9/(1 - x^3*A(x))^4 + x^16/(1 + x^4*A(x))^5 + x^25/(1 - x^5*A(x))^6 + x^36/(1 + x^6*A(x))^7 + ...
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PROG
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(PARI) /* Using: 1 = Sum_{n>=0} ( x^n + (-1)^n*A(x) )^n */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=1, #A, (x^m + (-1)^m*x*Ser(A))^m ), #A); ); polcoeff(A, n)}
for(n=1, 35, print1(a(n), ", "))
(PARI) /* Faster: 1 = Sum_{n>=0} x^(n^2) / (1 + (-x)^n*A(x))^(n+1) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, sqrtint(#A+1), x^(m^2)/(1 + (-x)^m*x*Ser(A))^(m+1) ), #A); ); polcoeff(A, n)}
for(n=1, 35, 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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