%I #76 Dec 21 2020 18:04:52

%S 1,3,75,4683,545835,102247563,28091567595,10641342970443,

%T 5315654681981355,3385534663256845323,2677687796244384203115,

%U 2574844419803190384544203,2958279121074145472650648875,4002225759844168492486127539083,6297562064950066033518373935334635,11403568794011880483742464196184901963

%N E.g.f.: Sum_{n>=0} exp(n^2*x) / 2^(n+1).

%C a(n) == 3 (mod 72) for n>0.

%C Conjectures from _Federico Provvedi_, Nov 07 2020: (Start)

%C For n>1, a(n+1) - a(n) == 0 (mod m) if and only if m divides 288.

%C This sequence is a periodic sequence modulo m, and if m is the k-th prime, the periods of {a(n)} over k-th prime is the sequence of the number of nonzero quadratic residues modulo k-th prime, for all k>0.

%C Example: k=9, m = prime(9) = 23, for n>0, {a(n) mod 23} generates a period of 11 elements {3, 6, 14, 22, 5, 3, 10, 2, 4, 5, 0}, hence A130290(9) = 11

%C (End)

%H Vaclav Kotesovec, <a href="/A249938/b249938.txt">Table of n, a(n) for n = 0..200</a>

%H Kwang-Wu Chen, <a href="https://arxiv.org/abs/1704.05636">Multinomial Sum Formulas of Multiple Zeta Values</a>, arXiv:1704.05636 [math.NT], 2017.

%F E.g.f.: (2 - cosh(x)) / (5 - 4*cosh(x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.

%F a(n) = Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>=0.

%F a(n) = A000670(2*n), where A000670 is the Fubini numbers.

%F a(n) ~ (2*n)! / (2 * (log(2))^(2*n+1)). - _Vaclav Kotesovec_, May 04 2015

%F a(n) = Sum_{p=1..k, q=1..k} Stirling2(k,p)*Stirling2(k,q)*p!*q!*A008288(p, q) for n>1, where A008288 are the Delannoy numbers. See Chen link. - _Michel Marcus_, Apr 20 2017

%F a(n) = Sum_{k>=0} k^(2*n) / 2^(k + 1). - _Ilya Gutkovskiy_, Dec 19 2019

%F a(n) = -Polylog(-2*n, 2) / 2. - _Federico Provvedi_, Nov 07 2020

%F a(n) = Phi(1/2, -2*n, 0), where Phi(z,s,a) is the Hurwitz-Lerch Zeta transcendental function. - _Federico Provvedi_, Nov 11 2020

%e E.g.f.: A(x) = 1 + 3*x + 75*x^2/2! + 4683*x^3/3! + 545835*x^4/4! +...

%e where the e.g.f. equals the infinite series:

%e A(x) = 1/2 + exp(x)/2^2 + exp(4*x)/2^3 + exp(9*x)/2^4 + exp(16*x)/2^5 + exp(25*x)/2^6 + exp(36*x)/2^7 + exp(49*x)/2^8 +...

%t Table[Sum[k! * StirlingS2[2*n, k],{k,0,2*n}],{n,0,20}] (* _Vaclav Kotesovec_, May 04 2015 *)

%t Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; a[n_] := Fubini[2n, 1]; a[0] = 1; Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Mar 30 2016 *)

%t Table[-PolyLog[-2*n, 2] / 2, {n, 0, 48}] (* _Federico Provvedi_, Nov 07 2020 *)

%t HurwitzLerchPhi[1/2, -2*Range[0,48], 0] / 2 (* _Federico Provvedi_, Nov 11 2020 *)

%t -HurwitzLerchPhi[2, -2*Range[0, 48], 1] (*_Federico Provvedi_,Nov 11 2020*)

%o (PARI) /* E.g.f.: Sum_{n>=0} exp(n^2*x)/2^(n+1) */

%o \p100 \\ set precision

%o {a(n) = round( n!*polcoeff(sum(m=0, 600, exp(m^2*x +x*O(x^n))/2^(m+1)*1.), n) )}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* E.g.f.: (2 - cosh(x)) / (5 - 4*cosh(x)): */

%o {a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (2 - cosh(X)) / (5 - 4*cosh(X)) , 2*n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* Formula for a(n): */

%o {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}

%o {a(n) = sum(k=0, 2*n, k! * Stirling2(2*n, k) )}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A249941, A247082, A250914, A250915, A000670.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 20 2014