A255881 Expansion of exp( Sum_{n >= 1} A000364(n)*x^n/n ).

1, 1, 3, 23, 371, 10515, 461869, 28969177, 2454072147, 269732425859, 37312477130105, 6342352991066661, 1299300852841580893, 315702973949640373933, 89765549161833322593411, 29526682496433138896248775, 11124674379405792463701519059

Offset: 0

Author

Peter Bala, Mar 09 2015

Keywords

• nonn
• easy

Comments

A000364(n) = (-1)^n*2^(2*n)*Euler(2*n,1/2), where E(n,x) is the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255882 (k = 3), A255883(k = 4) and A255884 (k = 6).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200
• E. W. Weisstein, Euler Polynomial

Crossrefs

Cf. A000364, A188514, A255882, A255883, A255884.

Programs

Maple

#A255881
k := 2:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 16)): seq(coeftayl(%, x = 0, n), n = 0 .. 16);

Mathematica

A000364:= Table[Abs[EulerE[2 n]], {n, 0, 80}]; a:= With[{nmax = 70}, CoefficientList[Series[Exp[Sum[A000364[[k + 1]]*x^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

Formula

O.g.f.: exp( x + 5*x^2/2 + 61*x^3/3 + 1385*x^4/4 + ... ) = 1 + x + 3*x^2 + 23*x^3 + 371*x^4 + ....

a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*2^(2*k)*E(2*k,1/2)*a(n-k).

a(n) ~ 2^(4*n + 3) * n^(2*n - 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019