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A216410
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E.g.f.: Series_Reversion( 2*Sw(x/2) ) where Sw(x) = Sum_{n>=0} (-1)^n*(2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
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0
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1, 4, 79, 3872, 357021, 53366688, 11788384035, 3613002977280, 1467889838452377, 763713003999744000, 495264178234423963575, 391720087063508887535616, 371190938737957616525807925, 415169544652854511226963558400, 541213248030886833323809041196875
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) satisfies: Sum_{n>=0} (-1)^n*(n+1)^(2*n) * A(x)^(2*n+1)/(2*n+1)! = x.
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EXAMPLE
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E.g.f.: A(x) = x + 4*x^3/3! + 79*x^5/5! + 3872*x^7/7! + 357021*x^9/9! +...
such that A(2*Sw(x/2)) = x, where
2*Sw(x/2) = x - 4*x^3 + 81*x^5 - 4096*x^7 + 390625*x^9 - 60466176*x^11 +...+ (-1)^n*(n+1)^(2*n)*x^(2*n+1)/(2*n+1)! +...
Related expansions:
Sw(x) = x - 16*x^3/3! + 1296*x^5/5! - 262144*x^7/7! + 100000000*x^9/9! -+...+ (-1)^n*(2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...
Cw(x) = 1 - 3*x^2/2! + 125*x^4/4! - 16807*x^6/6! + 4782969*x^8/8! -+...+ (-1)^n*(2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...
where Cw(x) + I*Sw(x) = LambertW(-I*x)/(-I*x).
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PROG
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(PARI) {a(n)=local(A=x); A=serreverse(sum(m=0, n\2, (-1)^m*(m+1)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 21, print1(a(2*n-1), ", ")) \\ print only odd-indexed terms
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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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