A014182 Expansion of e.g.f. exp(1-x-exp(-x)).

1, 0, -1, 1, 2, -9, 9, 50, -267, 413, 2180, -17731, 50533, 110176, -1966797, 9938669, -8638718, -278475061, 2540956509, -9816860358, -27172288399, 725503033401, -5592543175252, 15823587507881, 168392610536153, -2848115497132448, 20819319685262839

Offset

0,5

Comments

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

The 10-adic sum: B(n) = Sum_{k>=0} k^n*k! simplifies to: B(n) = A014182(n)*B(0) + A014619(n) for n>=0, where B(0) is the 10-adic sum of factorials (A025016); a result independent of base. - Paul D. Hanna, Aug 12 2006

Equals row sums of triangle A143987 and (shifted) = right border of A143987. [Gary W. Adamson, Sep 07 2008]

From Gary W. Adamson, Dec 31 2008: (Start)

Equals the eigensequence of the inverse of Pascal's triangle, A007318.

Binomial transform shifts to the right: (1, 1, 0, -1, 1, 2, -9, ...).

Double binomial transform = A109747. (End)

Convolved with A154107 = A000110, the Bell numbers. - Gary W. Adamson, Jan 04 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f.: exp(1-x-exp(-x)).

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n+1,k+1). - Paul D. Hanna, Aug 12 2006

A000587(n+1) = -a(n). - Michael Somos, May 12 2012

G.f.: 1/x/(U(0)-x) -1/x where U(k)= 1 - x + x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2012

G.f.: 1/(U(0) - x) where U(k) = 1 + x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2012

G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+k*x+x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013

G.f.: G(0)/(1+x)-1 where G(k) = 1 + 1/(1 + k*x - x*(1+k*x)*(1+k*x+x)/(x*(1+k*x+x) + (1+k*x+2*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2013

G.f.: S-1 where S = Sum_{k>=0} (2 + x*k)*x^k/Product_{i=0..k} (1+x+x*i). - Sergei N. Gladkovskii, Feb 09 2013

G.f.: G(0)*x^2/(1+x)/(1+2*x) + 2/(1+x) - 1 where G(k) = 1 + 2/(x + k*x - x^3*(k+1)*(k+2)/(x^2*(k+2) + 2*(1+k*x+3*x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2013

G.f.: 1/(x*Q(0)) -1/x, where Q(k) = 1 - x/(1 + (k+1)*x/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013

G.f.: G(0)/(1-x)/x - 1/x, where G(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (x*k + 1 - x)*(x*k + 1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2014

G.f.: (1 - Sum_{k>0} k * x^k / ((1 + x) * (1 + 2*x) + ... (1 + k*x))) / (1 - x). - Michael Somos, Nov 07 2014

a(n) = exp(1) * (-1)^n * Sum_{k>=0} (-1)^k * (k + 1)^n / k!. - Ilya Gutkovskiy, Dec 20 2019

Example

G.f. = 1 - x^2 + x^3 + 2*x^4 - 9*x^5 + 9*x^6 + 50*x^7 - 267*x^8 + 413*x^9 + ...

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[1-x-Exp[-x]], {x, 0, nn}], x] Range[0, nn]!]  (* Harvey P. Dale, Jan 15 2012 *)
a[ n_] := SeriesCoefficient[ (1 - Sum[ k / Pochhammer[ 1/x + 1, k], {k, n}]) / (1 - x), {x, 0, n} ]; (* Michael Somos, Nov 07 2014 *)

Prog

(PARI) {a(n)=sum(j=0, n, (-1)^(n-j)*Stirling2(n+1, j+1))}
{Stirling2(n, k)=(1/k!)*sum(i=0, k, (-1)^(k-i)*binomial(k, i)*i^n)} \\ Paul D. Hanna, Aug 12 2006
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - x - exp( -x + x * O(x^n))), n))} /* Michael Somos, Mar 11 2004 */
(Sage)
def A014182_list(len):  # len>=1
    T = [0]*(len+1); T[1] = 1; R = [1]
    for n in (1..len-1):
        a, b, c = 1, 0, 0
        for k in range(n, -1, -1):
            r = a - k*b - (k+1)*c
            if k < n : T[k+2] = u;
            a, b, c = T[k-1], a, b
            u = r
        T[1] = u; R.append(u)
    return R
A014182_list(27)  # Peter Luschny, Nov 01 2012

Crossrefs

Essentially same as A000587. See also A014619.

Cf. A025016.

Cf. A143987, A109747, A154107, A000110.

Sequence in context: A242064 A109322 A000587 * A293037 A131463 A065644

Adjacent sequences: A014179 A014180 A014181 * A014183 A014184 A014185

Keyword

sign,easy,nice

Author

Noam D. Elkies

Status

approved