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A305134 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1). 2
1, 6, 106, 9798, 2042986, 721198086, 378754904746, 274462194065478, 261211828432706026, 315282684090141417606, 470124979835875652863786, 848422945353825106452994758, 1822526603267557240862350671466, 4596139606368556055825161023870726, 13448584326250762088160567798167642026, 45199506338787031550197525974862852621638 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..50

FORMULA

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:

(1) 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).

(2) 1 = Sum_{n>=0} 2^n * exp(n^2*x) / (2 + exp(n*x) * A(x))^(n+1).

EXAMPLE

E.g.f.: A(x) = 1 + 6*x + 106*x^2/2! + 9798*x^3/3! + 2042986*x^4/4! + 721198086*x^5/5! + 378754904746*x^6/6! + 274462194065478*x^7/7! + 261211828432706026*x^8/8! + 315282684090141417606*x^9/9! + 470124979835875652863786*x^10/10! + ...

such that

1 = 1/2  +  (2*exp(x) - A(x))/2^2  +  (2*exp(2*x) - A(x))^2/2^3  +  (2*exp(3*x) - A(x))^3/2^4  +  (2*exp(4*x) - A(x))^4/2^5  +  (2*exp(5*x) - A(x))^5/2^6 + ...

Also,

1 = 1/(2 + A(x))  +  2*exp(x)/(2 + exp(x)*A(x))^2  +  2^2*exp(4*x)/(2 + exp(2*x)*A(x))^3  +  2^3*exp(9*x)/(2 + exp(3*x)*A(x))^4  +  2^4*exp(16*x)/(2 + exp(4*x)*A(x))^5  +  2^5*exp(25*x)/(2 + exp(5*x)*A(x))^6  + ...

RELATED SERIES.

log(A(x)) = 6*x + 70*x^2/2! + 8322*x^3/3! + 1812142*x^4/4! + 657412530*x^5/5! + 351254035150*x^6/6! + 257586196964082*x^7/7! + 247297892785673422*x^8/8! + 300478711708843324530*x^9/9! + 450397140484880214948430*x^10/10! + ...

CROSSREFS

Cf. A304640, A301436.

Sequence in context: A213464 A213465 A214349 * A083432 A006453 A251667

Adjacent sequences:  A305131 A305132 A305133 * A305135 A305136 A305137

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 29 2018

STATUS

approved

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Last modified July 10 15:45 EDT 2020. Contains 335577 sequences. (Running on oeis4.)