A320961 The exponential limit of (-x)!, rounded to the nearest integer.

1, 1, 4, 27, 353, 6128, 145159, 4402407, 166608593, 7666343436, 420646243820, 27079750092637, 2018074017351900, 172131994564410026, 16641769389384512884, 1808431867178308597550, 219272140061011055068448, 29473880023661693302772550, 4366902281695075479226089449

Offset

0,3

Comments

The exponential limit of a function is defined in A320956. Applied to (-x)! Maple returns a sequence of sums of Zeta values, powers of Pi, powers of Euler's gamma, etc.. The sequence starts: 1, gamma, (1/3)*Pi^2 + 2* gamma^2, 10*Zeta(3) + (5/2)*Pi^2*gamma + 5*gamma^3, ... These sums, rounded to the nearest integer, give the sequence.

Links

Table of n, a(n) for n=0..18.

Maple

explim := (len, f) -> seq(combinat:-bell(n)*((D@@n)(f))(0), n=0..len):
explim(18, x -> (-x)!): map(round, [evalf(%, 46)]);

Mathematica

m = 19; CoefficientList[(-x)!+O[x]^m, x]*Range[0, m-1]!*BellB[Range[0, m-1]] // Round (* Jean-François Alcover, Jul 21 2019 *)

Crossrefs

Cf. A320956, A000142.

The exponential limit of other functions: A320955 (exp), A320962 (log(x+1)), A320958 (arcsin), A320959 (arctanh).

Sequence in context: A221411 A304654 A203202 * A239726 A058155 A334527

Adjacent sequences: A320958 A320959 A320960 * A320962 A320963 A320964

Keyword

nonn

Author

Peter Luschny, Nov 07 2018

Status

approved