A255883 Expansion of exp( Sum_{n >= 1} A000281(n)*x^n/n ).

1, 3, 33, 1011, 65985, 7536099, 1329205857, 334169853267, 113370124235649, 49880529542872515, 27614111852126579361, 18782012442066306225843, 15394836674855296870428993, 14965462261283347594195897251, 17023467576167762236198869304545, 22400927665017118737825435362462739

Offset

0,2

Comments

A000281(n) =(-1)^n*4^(2*n)*E(2*n,1/4), where E(n,x) denotes 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 A255881 (k = 2), A255882(k = 3) and A255884 (k = 6).

Links

G. C. Greubel, Table of n, a(n) for n = 0..200

E. W. Weisstein, Euler Polynomial

Formula

O.g.f.: exp( 3*x + 57*x^2/2 + 2763*x^3/3 + 250737*x^4/4 + ... ) = 1 + 3*x + 33*x^2 + 1011*x^3 + 65985*x^4 + ....

a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*4^(2*k)*E(2*k,1/4)*a(n-k).

a(n) ~ 2^(6*n + 5/2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

Maple

#A255883
k := 4:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

Mathematica

A000281:= With[{nn = 200}, Take[CoefficientList[Series[Cos[x]/Cos[2 x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A000281[[k + 1]]*x^(k)/(k), {k, 1, 85}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}]  (* G. C. Greubel, Aug 26 2018 *)

Crossrefs

Cf. A000281, A188514, A255881, A255882, A255884

Sequence in context: A055549 A086894 A255930 * A215948 A012487 A188387

Adjacent sequences: A255880 A255881 A255882 * A255884 A255885 A255886

Keyword

nonn,easy

Author

Peter Bala, Mar 09 2015

Status

approved