A255882 Expansion of exp( Sum_{n >= 1} A210657(n)*(-x)^n/n ).

1, 2, 13, 224, 8170, 522716, 51749722, 7309866728, 1394040714169, 344865267322010, 107361980072755261, 41067497940750566312, 18931745446455458282248, 10350955324610065848650384, 6622526747212249020075069880, 4901565185965701578921602882976

Offset: 0

Author

Peter Bala, Mar 09 2015

Keywords

• nonn
• easy

Comments

A210657(n) = 3^(2*n)*E(2*n,1/3), where E(n,x) is the n-th Euler polynomial. In general it appears that when is k 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), 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. A188514, A210657, A255881, A255883, A255884.

Programs

Maple

#A255882
k := 3:
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

A210657[n_]:= 9^n EulerE[2 n, 1/3]; a:= With[{nmax = 80}, CoefficientList[Series[Exp[Sum[A210657[k]*(-x)^(k)/(k), {k, 1, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 51}] (* G. C. Greubel, Aug 26 2018 *)

Formula

O.g.f.: exp( 2*x + 22*x^2/2 + 602*x^3/3 + 30742*x^4/4 + ... ) = 1 + 2*x + 13*x^2 + 224*x^3 + 8170*x^4 + ....

a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*3^(2*k)*E(2*k,1/3)*a(n-k).

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