A255895 - OEIS

%I #12 Nov 05 2021 06:29:32

%S 1,5,43,635,15811,626543,36670645,2968583909,316359049075,

%T 42804896729263,7156058577755345,1447112885406569905,

%U 347972935518441794285,98096607566761160791345,32036179679728008237066395,11996707788374286511611298267,5105803366761640532801681583635

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

%C 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.

%F a(n) ~ 2^(4*n + 7) * n^(2*n + 3/2) / (exp(2*n) * Pi^(2*n + 5/2)). - _Vaclav Kotesovec_, Nov 05 2021

%e O.g.f.: A(x) = 1 + 5*x + 43*x^2 + 635*x^3 + 15811*x^4 + 626543*x^5 +...

%e where

%e 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 +...

%t 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 *)

%o (PARI) /* By the Ordinary Generating Function: */

%o {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)}

%o {a(n)=local(A=1);A=exp(sum(m=1,n,A000364(m+1)*x^m/m) +x*O(x^n));polcoeff(A,n)}

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

%Y Cf. A255881, A000364.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 09 2015