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

Author

Peter Luschny, Nov 07 2018

Keywords

• nonn

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.

Crossrefs

Cf. A320956, A000142.

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

Programs

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 *)