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A245308
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E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * ( d/dx x*A(x)^n ) / A(x)^n.
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3
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1, 1, 3, 19, 205, 3501, 90271, 3357103, 171841209, 11598601465, 996140770651, 105829573610091, 13602095395648453, 2077762791361106149, 371766799417828843575, 76978381709312988826951, 18256702588619162109630961, 4915636696257611754342845553, 1491009565882345791444427756339
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f. satisfies:
(1) A(x) = exp(x) * (1 + x^2*A'(x)/A(x)).
(2) A(x) = exp( Sum_{n>=1} A245119(n+1)*x^n/n ).
a(n) ~ c * (n!)^2 / n, where c = BesselJ(1,2) = 0.5767248077568733872... . - Vaclav Kotesovec, Jul 22 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 205*x^4/4! + 3501*x^5/5! +...
such that A(x) = exp(x) * (1 + x^2*A'(x)/A(x)) where
1 + x^2*A'(x)/A(x) = 1 + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 2640*x^5/5! + 72000*x^6/6! + 2792160*x^7/7! + 147329280*x^8/8! +...
RELATED SERIES.
The e.g.f. equals the product of exp(x) and an integer series (A245119):
exp(-x)*A(x) = 1 + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 100*x^6 + 554*x^7 + 3654*x^8 + 28014*x^9 + 244572*x^10 + 2392042*x^11 + 25877610*x^12 +...+ A245119(n)*x^n +...
The logarithmic derivative of the e.g.f. is an integer series:
A'(x)/A(x) = 1 + 2*x + 6*x^2 + 22*x^3 + 100*x^4 + 554*x^5 + 3654*x^6 + 28014*x^7 + 244572*x^8 + 2392042*x^9 + 25877610*x^10 +...+ A245119(n+2)*x^n +...
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*deriv(x*A^m)/A^m/m!+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*(1+m*x*A'/A)/m!+x*O(x^n))); n!*polcoeff(A, n)}
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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