|
%I #23 Jul 09 2019 00:05:53
%S 1,0,0,-4,6,-8,0,-12,-8400,40304,0,-20,-12640320,-24,0,-50854003228,
%T 523588665600,-32,0,-36,-47601422668800,-351419178958848040,0,-44,
%U -2657154018657032294400,60321372390731612159952,0,-3140712271670490316800052,-542080978696857600000,-56,0,-60,-348600503774477204939323514880000,-50178168859710075748378214400064,0,-1161795736961405183891091750912000068
%N E.g.f.: 2*cosh(1) - Sum_{n>=0} (1 + x^n)^n * exp(-x^n) / n!.
%C More generally, the following sums are equal:
%C (1) 2*cosh(p*r) - Sum_{n>=0} (q^n + p)^n * exp(-p*q^n*r) * r^n/n!,
%C (2) 2*cosh(p*r) - Sum_{n>=0} (q^n - p)^n * exp(+p*q^n*r) * r^n/n!,
%C under suitable conditions; here, p = 1, q = x, r = 1.
%H Paul D. Hanna, <a href="/A326426/b326426.txt">Table of n, a(n) for n = 0..520</a>
%F E.g.f.: 2*cosh(1) - Sum_{n>=0} (1 + x^n)^n * exp(-x^n) / n!.
%F E.g.f.: 2*cosh(1) - Sum_{n>=0} (-1)^n * (1 - x^n)^n * exp(x^n) / n!.
%F a(4*n+2) = 0 for n >= 0.
%e E.g.f.: A(x) = 1 - 4*x^3/3! + 6*x^4/4! - 8*x^5/5! - 12*x^7/7! - 8400*x^8/8! + 40304*x^9/9! - 20*x^11/11! - 12640320*x^12/12! - 24*x^13/13! - 50854003228*x^15/15! + 523588665600*x^16/16! - 32*x^17/17! - 36*x^19/19! - 47601422668800*x^20/20! - 351419178958848040*x^21/21! - 44*x^23/23! - 2657154018657032294400*x^24/24! + 60321372390731612159952*x^25/25! + ...
%e such that
%e A(x) = 2*cosh(1) - (exp(-1) + (1+x)*exp(-x) + (1+x^2)^2*exp(-x^2)/2! + (1+x^3)^3*exp(-x^3)/3! + (1+x^4)^4*exp(-x^4)/4! + (1+x^5)^5*exp(-x^5)/5! + ...)
%e also
%e A(x) = 2*cosh(1) - (exp(1) - (1-x)*exp(x) + (1-x^2)^2*exp(x^2)/2! - (1-x^3)^3*exp(x^3)/3! + (1-x^4)^4*exp(x^4)/4! - (1-x^5)^5*exp(x^5)/5! +- ...).
%e RELATED SERIES.
%e Below we illustrate the following identity at specific values of x:
%e 2*cosh(1) - Sum_{n>=0} (1 + x^n)^n * exp(-x^n) / n! = 2*cosh(1) - Sum_{n>=0} (-1)^n * (1 - x^n)^n * exp(x^n) / n!.
%e (1) At x = 1/2, the following sums are equal
%e S1 = 2*cosh(1) - Sum_{n>=0} (1 + 1/2^n)^n * exp(-1/2^n) / n!,
%e S1 = 2*cosh(1) - Sum_{n>=0} (1 - 1/2^n)^n * exp(1/2^n) * (-1)^n / n!,
%e where S1 = 0.92958560659999209901292951379518757326414006011141201017999...
%e (2) At x = 1/e, the following sums are equal
%e S2 = (e + 1/e) - Sum_{n>=0} (1 + 1/e^n)^n * e^(-1/e^n) / n!,
%e S2 = (e + 1/e) - Sum_{n>=0} (1 - 1/e^n)^n * e^(1/e^n) * (-1)^n / n!,
%e where S2 = 0.97087981067504743501007428564669808856438405820315797815416...
%e Compare the sums for S2 to the dual identity (cf. A326428)
%e S3 = (e + 1/e) - Sum_{n>=0} (1 - 1/e^n)^n * e^(-1/e^n) / n!,
%e S3 = (e + 1/e) - Sum_{n>=0} (1 + 1/e^n)^n * e^(1/e^n) * (-1)^n / n!,
%e where S3 = 1.77058869998597108214121594912015480934002883992351165553600...
%o (PARI) /* quick informal code for the initial 120 terms */
%o \p500
%o Vec(round(serlaplace( 2*cosh(1) - sum(n=0, 500, (1 + x^n +O(x^121))^n * exp(-x^n +O(x^121)) * 1./n! ) )))
%Y Cf. A326425, A326427, A326428.
%K sign
%O 0,4
%A _Paul D. Hanna_, Jul 03 2019
| 