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A245308 E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * ( d/dx x*A(x)^n ) / A(x)^n. 3
1, 1, 3, 19, 205, 3501, 90271, 3357103, 171841209, 11598601465, 996140770651, 105829573610091, 13602095395648453, 2077762791361106149, 371766799417828843575, 76978381709312988826951, 18256702588619162109630961, 4915636696257611754342845553, 1491009565882345791444427756339 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

FORMULA

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

EXAMPLE

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 +...

PROG

(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), ", "))

CROSSREFS

Cf. A245119.

Sequence in context: A201827 A230321 A108993 * A182956 A052886 A180563

Adjacent sequences:  A245305 A245306 A245307 * A245309 A245310 A245311

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 21 2014

STATUS

approved

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Last modified May 23 13:34 EDT 2022. Contains 353975 sequences. (Running on oeis4.)