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A295131
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G.f. A(x) satisfies: Product_{n=-oo..+oo} [1 + (-A(x))^n * (1 - (-A(x))^n)^n] = 2*x.
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2
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1, 2, 5, 16, 64, 291, 1397, 6875, 34450, 175615, 909627, 4776799, 25371720, 136033988, 735186170, 4000684739, 21902649976, 120555380997, 666732745113, 3703195116792, 20647931305136, 115529616213809, 648470586919557, 3650468031125360, 20604592863012251, 116585628797051735, 661165592229317701, 3757399793346622340
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OFFSET
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1,2
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COMMENTS
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Compare g.f. to: Sum_{n=-oo..+oo} y^n*(1 - y^n)^n = 0.
Limit a(n+1)/a(n) seems to be near 6.
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LINKS
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FORMULA
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G.f. A = A(x) satisfies P(x) * Q(x) = 2*x, where
P(x) = Product_{n>=0} [ 1 + (-A)^n * (1 - (-A)^n)^n ],
Q(x) = Product_{n>=1} [ 1 + (-1)^n * A^(n^2-n) / (1 - (-A)^n)^n ].
a(n) ~ c * d^n / n^(3/2), where d = 6.0047759432392225596564587487484652... and c = 0.0874195097648934898527627212249... - Vaclav Kotesovec, Nov 16 2017
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 5*x^3 + 16*x^4 + 64*x^5 + 291*x^6 + 1397*x^7 + 6875*x^8 + 34450*x^9 + 175615*x^10 + 909627*x^11 + 4776799*x^12 + 25371720*x^13 + 136033988*x^14 + 735186170*x^15 + 4000684739*x^16 + 21902649976*x^17 + 120555380997*x^18 + 666732745113*x^19 + 3703195116792*x^20 +...
such that A = A(x) satisfies P(x) * Q(x) = 2*x, where
P(x) = 2 * (1 - A*(1 + A)) * (1 + A^2*(1 - A^2)^2) * (1 - A^3*(1 + A^3)^3) * (1 + A^4*(1 - A^4)^4) * (1 - A^5*(1 + A^5)^5) * (1 + A^6*(1 - A^6)^6) *...
Q(x) = (1 - 1/(1 + A)) * (1 + A^2/(1 - A^2)^2) * (1 - A^6/(1 + A^3)^3) * (1 + A^12/(1 - A^4)^4) * (1 - A^20/(1 + A^5)^5) *...
Explicitly,
P(x) = 2 - 2*x - 4*x^2 - 14*x^3 - 58*x^4 - 252*x^5 - 1128*x^6 - 5228*x^7 - 25136*x^8 - 124758*x^9 - 634418*x^10 - 3284110*x^11 - 17232510*x^12 - 91422822*x^13 - 489573462*x^14 - 2643019882*x^15 - 14369975500*x^16 - 78614984310*x^17 - 432443167014*x^18 - 2390345670878*x^19 - 13270200493326*x^20 - 73959001618152*x^21 - 413656943495424*x^22 - 2321059075319620*x^23 - 13061936625432620*x^24 +...
Q(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 254*x^6 + 1223*x^7 + 6013*x^8 + 30189*x^9 + 154503*x^10 + 803559*x^11 + 4233992*x^12 + 22546635*x^13 + 121133264*x^14 + 655775428*x^15 + 3573885263*x^16 + 19591983934*x^17 + 107964357617*x^18 + 597725549532*x^19 + 3323033064351*x^20 + 18543921914138*x^21 + 103836577675255*x^22 + 583242599835564*x^23 + 3285362554801168*x^24 + 18554604800241247*x^25 +...
where Q(x) = 2*x / P(x).
The series reversion of g.f. A(x) begins:
Series_Reversion(A(x)) = x - 2*x^2 + 3*x^3 - 6*x^4 + 7*x^5 - 11*x^6 + 15*x^7 - 20*x^8 + 19*x^9 - 22*x^10 + 32*x^11 - 27*x^12 + 20*x^13 + 8*x^14 - 6*x^15 + 27*x^16 - 47*x^17 + 124*x^18 +...+ -A293602(n)/2*x^n +...
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PROG
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(PARI) {a(n) = my(A=[1, 0]); for(i=1, n, A = concat(A, 0); G = -x*Ser(A); A[#A-1] = -Vec(prod(m=-#A-1, #A+1, 1 + G^m*(1 - G^m)^m ))[#A-1]/2); A[n]}
for(n=1, 40, 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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