A317340 -id:A317340 - 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.

A317349 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n = 1.

Original entry on oeis.org

1, 1, 2, 7, 42, 372, 4269, 59047, 946557, 17175289, 347208299, 7730688884, 187911183701, 4951155672353, 140575561645293, 4279249948000903, 139050095246322895, 4804391579357016747, 175902340755219278039, 6803436418471129704925, 277202774381386656583959, 11868116969794805874111831

Offset: 0

Paul D. Hanna, Aug 02 2018

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 372*x^5 + 4269*x^6 + 59047*x^7 + 946557*x^8 + 17175289*x^9 + 347208299*x^10 + ...
such that
1 = 1  +  (1/A(x) - (1-x))  +  (1/A(x) - (1-x)^2)^2  +  (1/A(x) - (1-x)^3)^3  +  (1/A(x) - (1-x)^4)^4  +  (1/A(x) - (1-x)^5)^5  +  (1/A(x) - (1-x)^6)^6  +  (1/A(x) - (1-x)^7)^7  +  (1/A(x) - (1-x)^8)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - (1-x)^2)  +  (1/A(x) - (1-x)^3)^2  +  (1/A(x) - (1-x)^4)^3  +  (1/A(x) - (1-x)^5)^4  +  (1/A(x) - (1-x)^6)^5  +  (1/A(x) - (1-x)^7)^6  +  (1/A(x) - (1-x)^8)^7  +  (1/A(x) - (1-x)^9)^8  + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A317339, A317348, A317340, A317666, A317667 A317668.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( 1/Ser(A) - (1-x)^(m+1) )^m ) )[#A]/2 ); A[n+1]}
for(n=0,25, print1(a(n),", "))

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^n )^n.
(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n.
(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(n+1) )^n * (1-x)^(n+1).
(4) A(x)^2 = 2*A(x) * [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+1) )^n ] - [ Sum_{n>=0} (n+1) * ( 1/A(x) - (1-x)^(n+2) )^n ].
(5) A(x) = [ Sum_{n>=1} n*(n+1)/2 * (1-x)^(n+1) * ( 1/Ser(A) - (1-x)^(n+1) )^(n-1) ] / [ Sum_{n>=1} n^2 * (1-x)^n * ( 1/Ser(A) - (1-x)^n )^(n-1) ].
a(n) ~ 2^(log(2)/2 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 06 2018

cp's OEIS Frontend

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

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