A121886 -id:A121886 - cp's OEIS frontend

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

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

Offset: 1

Vaclav Kotesovec, Aug 09 2018

			3.161088653865428813830172202588132491726382774188556341627278...

Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.

Cf. A301584, A301585, A301586, A303056, A303057, A303654.

r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]

r=solve(r=.8,1,exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
(1+exp(1/r))*r^2 \\ Charles R Greathouse IV, Jun 15 2021

Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.

A220352 G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n^2).

Original entry on oeis.org

1, 1, 3, 16, 118, 1116, 12869, 175096, 2745726, 48756438, 967026762, 21188546616, 508286084222, 13249410224210, 372908807794347, 11270832179901016, 364083312029454453, 12518063823862065816, 456432182550333723335, 17591590487681007523476

Offset: 0

Paul D. Hanna, Dec 11 2012

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 118*x^4 + 1116*x^5 + 12869*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x/(1+x) + (2*x + x^2)^2/(1+x)^4 + (3*x + 3*x^2 + x^3)^3/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^4/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^5/(1+x)^25 +...
(2) A(x) = 1/(1+x) + (2*x + x^2)/(1+x)^4 + (3*x + 3*x^2 + x^3)^2/(1+x)^9 + (4*x + 6*x^2 + 4*x^3 + x^4)^3/(1+x)^16 + (5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)^4/(1+x)^25 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A220353, A121886, A187826.

{a(n)=local(q=1+x+x*O(x^n),A=1);A=sum(k=0,n,q^(-k^2)*(q^k-1)^k);polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

{a(n)=local(q=1+x+x*O(x^n),A=1);A=sum(k=1,n+1,q^(-k^2)*(q^k-1)^(k-1));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

G.f.: Sum_{n>=1} ((1+x)^n - 1)^(n-1) / (1+x)^(n^2).
a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 1.44832302735058524286860126583754380692... . - Vaclav Kotesovec, Nov 08 2014
In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1+log(2))/2)). - Vaclav Kotesovec, May 03 2015

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A317855 Decimal expansion of a constant related to the asymptotics of A122400.

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A220352 G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n^2).

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