A265953 -id:A265953 - cp's OEIS frontend

A265952 E.g.f.: Product_{k>=1} (1 + exp(x)*x^k).

1, 1, 4, 21, 136, 1045, 9396, 99379, 1161952, 14904873, 208925380, 3207759511, 53349017184, 950481426349, 17959336305652, 360094085423115, 7656053670162496, 172173524035504849, 4077253037751090948, 101263725993658291615, 2627592803013505930240

Offset: 0

Vaclav Kotesovec, Dec 19 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A265953.

nmax=20; CoefficientList[Series[Product[(1+E^x*x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

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

Original entry on oeis.org

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

A322612 Expansion of e.g.f. Product_{k>=1} 1/(1 + log(1 - x)*x^k).

Original entry on oeis.org

1, 0, 2, 9, 68, 490, 5184, 53928, 696352, 9545184, 147901680, 2437886880, 44593856064, 861936989472, 17988878376000, 398199273907680, 9386173867046400, 233068382185213440, 6117261434418069504, 168414066137504272896, 4867992707164288773120, 147081824197157871866880, 4641822165217412602183680

Offset: 0

Ilya Gutkovskiy, Dec 20 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..427

Cf. A227682, A265953, A322613.

seq(coeff(series(factorial(n)*mul((1+log(1-x)*x^k)^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Dec 21 2018

nmax = 22; CoefficientList[Series[Product[1/(1 + Log[1 - x] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1/(1 - x)]^d/d, {d, Divisors[k]}] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} log(1/(1 - x))^d/d ) * x^k).

a(n) ~ c * n! / r^n, where r = 0.74075364335169502373416717320773551326074821766... is the root of the equation r*log(1-r) = -1 and c = 1 / (r*(r/(1-r) - log(1-r)) * Product_{k>=2} (1 + log(1-r)*r^k) ) = 16.634865259935976898139371781860039862... - Vaclav Kotesovec, Dec 20 2018

A302399 Expansion of e.g.f. Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

Original entry on oeis.org

1, 1, 8, 63, 628, 7405, 103266, 1630195, 28812344, 561715353, 11971270270, 276322667071, 6867229990644, 182651988444133, 5174629835814362, 155498722020145995, 4938797154614179696, 165259917542803746097, 5809661798192528407542, 214032701720169039806551, 8244827039453943163648940

Offset: 0

Ilya Gutkovskiy, Apr 07 2018

nonn

			Product_{k>=1} 1/(1 - exp(x)*x^k)^k = 1 + x/1! + 8*x^2/2! + 63*x^3/3! + 628*x^4/4! + 7405*x^5/5! + 103266*x^6/6! + ...

Cf. A265952, A265953.

a:=series(mul(1/(1-exp(x)*x^k)^k,k=1..100),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[Product[1/(1 - Exp[x] x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

a(n) ~ c * n! / LambertW(1)^n, where c = 1/(1 + LambertW(1)) * Product_{j>=1} 1/(1 - LambertW(1)^j)^(j+1) = 115.50749040505570853455997830821388214033876679679... - Vaclav Kotesovec, Apr 07 2018

A347005 E.g.f.: Product_{k>=1} 1 / (1 - exp(x) * x^k / k!).

Original entry on oeis.org

1, 1, 5, 28, 205, 1856, 19964, 249005, 3535613, 56339884, 996009280, 19350090365, 409850078356, 9400728524669, 232154433941057, 6141705628777193, 173295665869432733, 5195039603196754564, 164890990869273983108, 5524278740902526776085, 194815729875439415542760

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Cf. A005651, A140585, A265953, A347006.

nmax = 20; CoefficientList[Series[Product[1/(1 - Exp[x] x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp( Sum_{k>=1} ( Sum_{d|k} exp(d*x) / (d * ((k/d)!)^d) ) * x^k ).
E.g.f.: Product_{k>=1} 1 / (1 - Sum_{j>=k} binomial(j,k) * x^j / j!).
a(n) ~ c * n! / ((1 + LambertW(1)) * LambertW(1)^n), where c = Product_{k>=2} (1/(1 - LambertW(1)^(k-1)/k!)) = 1.487589725380080111479849424209442083... - Vaclav Kotesovec, Aug 10 2021

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A265952 E.g.f.: Product_{k>=1} (1 + exp(x)*x^k).

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

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A322612 Expansion of e.g.f. Product_{k>=1} 1/(1 + log(1 - x)*x^k).

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A302399 Expansion of e.g.f. Product_{k>=1} 1/(1 - exp(x)*x^k)^k.

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A347005 E.g.f.: Product_{k>=1} 1 / (1 - exp(x) * x^k / k!).

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