A255895 O.g.f.: exp( Sum_{n>=1} A000364(n+1)*x^n/n ), where A000364 is the Euler numbers.

1, 5, 43, 635, 15811, 626543, 36670645, 2968583909, 316359049075, 42804896729263, 7156058577755345, 1447112885406569905, 347972935518441794285, 98096607566761160791345, 32036179679728008237066395, 11996707788374286511611298267, 5105803366761640532801681583635

Offset: 0

Author

Paul D. Hanna, Mar 09 2015

Keywords

• nonn

Comments

Inspired by Peter Bala's sequence A255881, which suggests the following conjecture: exp( Sum_{n>=1} A000364(n+m)*x^n/n ) is an integer series for integer m>=0.

Examples

			O.g.f.: A(x) = 1 + 5*x + 43*x^2 + 635*x^3 + 15811*x^4 + 626543*x^5 +...
where
log(A(x)) = 5*x + 61*x^2/2 + 1385*x^3/3 + 50521*x^4/4 + 2702765*x^5/5 + 199360981*x^6/6 + 19391512145*x^7/7 + 2404879675441*x^8/8 +...+ A000364(n+1)*x^n/n +...
		

Crossrefs

Cf. A255881, A000364.

Programs

Mathematica

Table[SeriesCoefficient[Exp[Sum[Abs[EulerE[2*k+2]]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)

PARI

/* By the Ordinary Generating Function: */
{A000364(n)=polcoeff(sum(m=0, n, (2*m)!/2^m * x^m/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)}
{a(n)=local(A=1);A=exp(sum(m=1,n,A000364(m+1)*x^m/m) +x*O(x^n));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ 2^(4*n + 7) * n^(2*n + 3/2) / (exp(2*n) * Pi^(2*n + 5/2)). - Vaclav Kotesovec, Nov 05 2021