A014619 Exponential generating function is -f(x) * Integral_{t = 0..x} exp(exp(-t) - 1) dt, where f(x) = exp(1 - x - exp(-x)) is the exponential generating function for A014182.

-1, 1, 1, -5, 5, 21, -105, 141, 777, -5513, 13209, 39821, -527525, 2257425, -41511, -70561285, 531862173, -1559180499, -8858267353, 147780183829, -936560917615, 1352130196615, 38710924110081, -487251979381019, 2846575686392251, 872653153712201

Offset

1,4

Links

Table of n, a(n) for n=1..26.

Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.

Branko Dragovich, Andrei Yu. Khrennikov, and Natasa Z. Misic, Summation of p-Adic Functional Series in Integer Points, arXiv:1508.05079 [math.NT], 2015.

B. Dragovich and N. Z. Misic, p-Adic invariant summation of some p-adic functional series, P-Adic Numbers, Ultrametric Analysis, and Applications, October 2014, Volume 6, Issue 4, pp 275-283.

Formula

E.g.f. A(x) = y satisfies y'' + y'(2-exp(-x)) + y = 0. - Michael Somos, Mar 11 2004

a(n) = Sum_{k = 0..n} (-1)^(n-k+1)*Stirling2(n+1, k+1)*A003422(k). - Vladeta Jovovic, Jan 06 2005

The sequence b(n) = (-1)^n*a(n) satisfies the recurrence: b(n) = -Sum_{i = 1..n} b(i-1)*C(n, i), b(0) = -1. - Ralf Stephan, Feb 24 2005

From Peter Bala, Mar 23 2024: (Start)

It appears that a(n) = Sum_{k = 1..n+1} binomial(n+1, k)*a(k). See Dragovich 2017, Table 1.

If true then the following hold: setting a(0) = -1 then

a(n) = Sum_{k = 1..n-1} (-1)^(n-k)*binomial(n-1, k-1)*a(k-1);

the o.g.f F(x) = - ( x/(1 + x)^2 + x^2/((1 + x)*(1 + 2*x)^2) + x^3/((1 + x)*(1 + 2*x)*(1 + 3*x)^2) + ... ) - Cf. A040027;

F(x) = - x/(1 + x)^2 + x/(1 + x)^2*F(x/(1 + x)). (End)

Mathematica

a[n_] := Sum[(-1)^(n - k + 1) * StirlingS2[n + 1, k + 1] * ((-1)^k * k! * Subfactorial[-k - 1] - Subfactorial[-1]), {k, 0, n}]; Table[a[n] // FullSimplify, {n, 1, 26}] (* Jean-François Alcover, Jan 09 2014, after Vladeta Jovovic *)
nmax = 25; Rest[CoefficientList[Series[E^(-E^(-x) - x) * (Gamma[0, -1] - Gamma[0, -E^(-x)]), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, May 10 2024 *)

Prog

(PARI) a(n)=local(A, B); if(n<0, 0, A=exp(-x+x*O(x^n)); B=exp(A-1); n!*polcoeff(-intformal(B)*A/B, n))

Crossrefs

Cf. A000670, A040027, A163940.

Sequence in context: A318960 A097336 A365104 * A205140 A097016 A183257

Adjacent sequences: A014616 A014617 A014618 * A014620 A014621 A014622

Keyword

sign

Author

Noam D. Elkies

Extensions

More terms from Jason Earls, Jun 28 2001

Status

approved