The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A366232 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A366232

Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(2*x*A(x)).
(history; published version)

#20 by Paul D. Hanna at Sun Oct 08 06:45:21 EDT 2023

STATUS

editing

approved

#19 by Paul D. Hanna at Sun Oct 08 06:45:15 EDT 2023

EXAMPLE

RELATED SERIES.

exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ... + A212722(n)*x^n/n! + ...

CROSSREFS

Cf. A365772 (dual), A212722 (exp(x*A(x)).

STATUS

approved

editing

#18 by Paul D. Hanna at Sun Oct 08 06:25:39 EDT 2023

STATUS

editing

approved

#17 by Paul D. Hanna at Sun Oct 08 06:25:37 EDT 2023

COMMENTS

Related identities which hold formally for all power Maclaurin series F(x):

FORMULA

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

STATUS

approved

editing

#16 by Bruno Berselli at Fri Oct 06 10:02:22 EDT 2023

STATUS

proposed

approved

#15 by Vaclav Kotesovec at Fri Oct 06 04:00:10 EDT 2023

STATUS

editing

proposed

#14 by Vaclav Kotesovec at Fri Oct 06 03:59:32 EDT 2023

FORMULA

a(n) ~ n^(n-1) * (1 + 2*LambertW(1/sqrt(2)))^(n + 3/2) / (sqrt(1 + LambertW(1/sqrt(2))) * 2^(n+2) * exp(n) * LambertW(1/sqrt(2))^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

#13 by Vaclav Kotesovec at Fri Oct 06 03:16:00 EDT 2023

MATHEMATICA

nmax = 20; A[_] = 0; Do[A[x_] = 1 + x*A[x] * E^(2*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)

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proposed

editing

#12 by Paul D. Hanna at Thu Oct 05 21:43:33 EDT 2023

STATUS

editing

proposed

#11 by Paul D. Hanna at Thu Oct 05 21:43:31 EDT 2023

FORMULA

(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-2)*x*A(x)) for all fixed positive nonnegative m.

STATUS

proposed

editing