A276231 -id:A276231 - cp's OEIS frontend

A115287 Decimal expansion of 1/(1+LambertW(1)).

6, 3, 8, 1, 0, 3, 7, 4, 3, 3, 6, 5, 1, 1, 0, 7, 7, 8, 5, 2, 2, 4, 0, 7, 3, 8, 5, 5, 1, 9, 8, 8, 0, 3, 1, 4, 4, 4, 3, 9, 3, 3, 8, 4, 1, 2, 8, 9, 0, 2, 7, 6, 4, 0, 4, 1, 9, 4, 8, 3, 1, 9, 3, 6, 5, 0, 3, 4, 2, 1, 0, 1, 0, 5, 6, 7, 6, 0, 0, 8, 3, 0, 4, 1, 0, 0, 1, 8, 5, 2, 5, 1, 0, 5, 2, 7, 4, 8, 3, 3, 1, 5, 7, 0, 9

Offset: 0

Eric W. Weisstein, Jan 19 2006

			0.63810374336511077852...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Cristina B. Corcino, Roberto B. Corcino and István Mező, Continued fraction expansions for the Lambert W function, Aequationes mathematicae, Vol. 93, No. 2 (2019), pp. 485-49.
Mathematics Stack Exchange, Interesting integral related to the Omega Constant/Lambert W Function, 2011.
Victor H. Moll, Some Questions in the Evaluation of Definite Integrals, MAA Short Course, San Antonio, TX, January 2006.
Eric Weisstein's World of Mathematics, Omega Constant.
Eric Weisstein's World of Mathematics, Definite Integral.

Cf. A006153, A030178, A097172, A097173, A097174, A265953, A276231, A302399.

RealDigits[1/(1 + ProductLog[1]), 10, 105] // First (* Jean-François Alcover, Feb 07 2013 *)

1/(1 + lambertw(1)) \\ G. C. Greubel, Nov 05 2017

Equals Integral_{x=-oo..oo} 1/(Pi^2 + (exp(x)-x)^2) dx (discovered by Victor Adamchik). - Amiram Eldar, Jul 04 2021

A276232 E.g.f. W(x)^W(x), where W(x) = LambertW(-x)/(-x) and satisfies W(x)^(1/W(x)) = exp(x).

Original entry on oeis.org

1, 1, 5, 37, 369, 4641, 70513, 1256361, 25689569, 592998049, 15254145441, 432741923769, 13422771397489, 451956633181041, 16418360947494353, 640101032269166281, 26659869619363530177, 1181426235219348415041, 55506506359841719631425, 2756055067645593490566489, 144211201693287134720633681, 7931567651481815767204102801

Offset: 0

Paul D. Hanna, Aug 24 2016

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 369*x^4/4! + 4641*x^5/5! + 70513*x^6/6! + 1256361*x^7/7! + 25689569*x^8/8! + 592998049*x^9/9! + 15254145441*x^10/10! +...
such that A(x) = W(x)^W(x).
Define W(x) = LambertW(-x)/(-x), where
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
satisfies W(x) = exp(x)^W(x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..384

Cf. A276231.

CoefficientList[Series[E^(LambertW[-x]^2/x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 26 2016 *)

{a(n) = my(A=1+x, W); W=serreverse(x*exp(-x +x^2*O(x^n)))/x; A = W^W; n!*polcoeff(A,n)}
for(n=0, 40, print1(a(n), ", "))

E.g.f.: exp( LambertW(-x)^2/x ).

a(n) ~ 2*exp(exp(1)+1)*n^(n-1). - Vaclav Kotesovec, Aug 26 2016

cp's OEIS Frontend

A115287 Decimal expansion of 1/(1+LambertW(1)).

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