A255900 Expansion of exp( Sum_{n >= 1} A000464(n-1)*x^n/n ).

1, 1, 6, 126, 6291, 581499, 86010084, 18599726484, 5532984567639, 2166830287921959, 1080602568966548022, 668603866168566179982, 502601850887023486736757, 451167540495577093495790397, 476690705981608679350490956032, 585587125411920596898761442409728

Offset

0,3

Comments

For similar results see A188514 and A255881.

A000464(n-1) = (-1)^n*4^(2*n-1)*E(2*n-1,1/4), where E(n,x) denotes the n-th Euler polynomial.

More generally, calculation suggests that for integer h and a nonzero integer k the expansion of exp ( Sum_{n >= 1} (2*k)^(2*n-1)*E(2*n-1,h/(2*k)) )*x^n/n has integer coefficients. This is the case h = 1 and k = 2.

Links

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

Formula

O.g.f.: exp( x + 11*x^2/2 + 361*x^3/3 + 24611*x^4/4 + ... ) = 1 + x + 6*x^2 + 126*x^3 + 6291*x^4 + ....

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

Maple

#A255900
k := 2:
exp(add((2*k)^(2*n-1)*euler(2*n-1, 1/(2*k))*(-x)^n/n, n = 1 .. 15)): seq(coeftayl(%, x = 0, n), n = 0 .. 15);

Mathematica

A000464:= With[{nn = 200}, Take[CoefficientList[Series[Sin[x]/Cos[2 x], {x, 0, nn}], x]*Range[0, nn-1]!, {2, -1, 2}]]; a:= With[{nmax = 50}, CoefficientList[Series[Exp[x + Sum[A000464[[k]]*x^(k)/(k), {k, 2, 75}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

Crossrefs

Cf. A000464, A188514, A255881.

Sequence in context: A370715 A004993 A237428 * A133792 A081623 A223210

Adjacent sequences: A255897 A255898 A255899 * A255901 A255902 A255903

Keyword

nonn,easy

Author

Peter Bala, Mar 10 2015

Status

approved