A025016 -id:A025016 - cp's OEIS frontend

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

Noam D. Elkies

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

			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 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mohammad Ghorbani, Mehdi Hassani, and Hossein Moshtagh, Moments and asymptotic expansion of derangement polynomials in terms of Touchard polynomials, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 4, 832-842. See pp. 834, 839.

Essentially same as A000587. See also A014619.
Cf. A025016.
Cf. A143987, A109747, A154107, A000110.

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 *)

{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

{a(n) = if( n<0, 0, n! * polcoeff( exp( 1 - x - exp( -x + x * O(x^n))), n))} /* Michael Somos, Mar 11 2004 */

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

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

A064733 Final digits of A005165(2n) for large n, read from right.

Original entry on oeis.org

9, 1, 0, 7, 7, 5, 5, 3, 6, 1, 6, 1, 2, 6, 3, 9, 3, 5, 6, 4, 4, 1, 2, 8, 8, 5, 7, 6, 3, 2, 6, 8, 6, 5, 5, 0, 1, 6, 0, 4, 8, 5, 9, 1, 6, 9, 4, 6, 3, 4, 0, 7, 3, 8, 1, 7, 6, 8, 1, 3, 1, 6, 8, 8, 1, 1, 9, 8, 9, 2, 8, 8, 8, 6, 0, 2, 9, 5, 3, 9, 9, 6, 6, 3, 2, 9, 9, 3, 6, 3, 4, 2, 4, 7, 9, 0, 7, 7, 4, 5, 4, 6, 4, 8, 6

Offset: 0

Vladeta Jovovic, Oct 17 2001

			A005165(20)=2317323290554617019; A005165(30)=256705485669535347568006115577019.

Index entries for sequences related to final digits of numbers

Cf. A005165, A025016.

More terms from David Wasserman, Aug 05 2002

A065356 Final digits of A065355(n) (in reverse order) for sufficiently large n.

Original entry on oeis.org

6, 8, 6, 9, 5, 0, 9, 7, 5, 9, 7, 0, 1, 7, 4, 3, 6, 6, 7, 5, 5, 3, 4, 4, 7, 4, 9, 0, 6, 9, 4, 9, 8, 6, 0, 4, 6, 7, 6, 5, 9, 1, 5, 0, 0, 2, 9, 8, 8, 7, 3, 1, 6, 2, 5, 1, 3, 1, 2, 5, 0, 2, 5, 2, 5, 7, 7, 0, 9, 9, 5, 6, 6, 9, 4, 3, 4, 1, 3, 4, 9, 9, 7, 3, 3, 4, 8, 4, 0, 2, 1, 1, 8, 3, 7, 9, 7, 1, 8, 7, 8, 6, 2, 3, 8

Offset: 0

Floor van Lamoen, Oct 31 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A025016, A064733, A064734.

{ for (n=0, 1000, f=1; while(f!%10^(n+1)>0, f+=1); x=sum(k=0, f, k!%10^(n+1)); a=(10*(x%10^(n+1)))\10^(n+1); if (n==0, a=3); write("b065356.txt", n, " ", 9 - a) ) } \\ Harry J. Smith, Oct 17 2009

a(n) = 9-A025016(n), n > 0. - Vladeta Jovovic, Nov 02 2001

a(60)-a(104) from Harry J. Smith, Oct 17 2009

A364128 Decimal expansion of a constant related to A053529 and A179973.

Original entry on oeis.org

4, 4, 3, 2, 3, 8, 9, 5, 4, 7, 3, 0, 9, 2, 8, 5, 0, 9, 4, 0, 7, 7, 7, 5, 1, 2, 0, 7, 2, 8, 3, 3, 1, 8, 5, 1, 5, 0, 2, 0, 7, 2, 1, 9, 2, 4, 3, 9, 1, 5, 3, 0, 8, 7, 0, 7, 7, 6, 2, 9, 2, 8, 7, 8, 5, 3, 4, 5, 9, 1, 5, 9, 1, 4, 4, 7, 8, 7, 3, 5, 9, 3, 2, 5, 5, 7, 6, 1, 1, 6, 9, 2, 9, 1, 3, 8, 2, 8, 7, 1, 6, 4, 8, 5, 8, 8

Offset: 0

Alois P. Heinz, Jul 09 2023

The digits of this constant are also the final digits of A179973(n) and of the n-th partial sum of A053529 for n -> oo, read from right.

			0.443238954730928509407775120728331851502072192439153087... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A004086, A025016, A053529, A055642, A179973.

Equals lim_{n->oo} A004086(A179973(n))/10^A055642(A179973(n)).

Equals lim_{n->oo} A004086(s(n))/10^A055642(s(n)) with s(n) = Sum_{j=0..n} A053529(j).

A064734 Final digits of A005165(2n+1) for large n, read from right.

Original entry on oeis.org

1, 8, 9, 2, 2, 4, 4, 6, 3, 8, 3, 8, 7, 3, 6, 0, 6, 4, 3, 5, 5, 8, 7, 1, 1, 4, 2, 3, 6, 7, 3, 1, 3, 4, 4, 9, 8, 3, 9, 5, 1, 4, 0, 8, 3, 0, 5, 3, 6, 5, 9, 2, 6, 1, 8, 2, 3, 1, 8, 6, 8, 3, 1, 1, 8, 8, 0, 1, 0, 7, 1, 1, 1, 3, 9, 7, 0, 4, 6, 0, 0, 3, 3, 6, 7, 0, 0, 6, 3, 6, 5, 7, 5, 2, 0, 9, 2, 2, 5, 4, 5, 3, 5, 1, 3

Offset: 0

Vladeta Jovovic, Oct 17 2001

			A005165(21)=48773618881154822981; A005165(31)=7966133168508387470157556764422981.

Index entries for sequences related to final digits of numbers

Cf. A005165, A025016, A064733.

More terms from David Wasserman, Aug 05 2002

A082648 Consider f(m) = Sum_{k=1..m} k! (A007489) when m is very large; a(n) = n-th digit from end.

Original entry on oeis.org

3, 1, 3, 0, 4, 9, 0, 2, 4, 0, 2, 9, 8, 2, 5, 6, 3, 3, 2, 4, 4, 6, 5, 5, 2, 5, 0, 9, 3, 0, 5, 0, 1, 3, 9, 5, 3, 2, 3, 4, 0, 8, 4, 9, 9, 7, 0, 1, 1, 2, 6, 8, 3, 7, 4, 8, 6, 8, 7, 4, 9, 7, 4, 7, 4, 2, 2, 9, 0, 0, 4, 3, 3, 0, 5, 6, 5, 8, 6, 5

Offset: 1

Alexander Adamchuk, May 15 2003

Apart from the first term, the same as A025016. - R. J. Mathar, Sep 17 2008

Since A007845 gives the smallest factorial having at least n trailing zeros, the first n digits read from the right are determined for m >= A007845(n) - 1. - Martin Renner, Feb 14 2021

			Sum_{k=1..30} k! = 274410818470142134209703780940313.
The last 7 digits in reverse order give us the first 7 terms of this sequence: 3,1,3,0,4,9,0.
From _Jon E. Schoenfield_, Feb 16 2021: (Start)
The table below shows the 12 least-significant digits of Sum_{k=1..m} k! converging to the first 12 terms of this sequence (in reverse order) as m increases:
.
    m  Sum_{k=1..m} k!  # corresponding digits
   --  ---------------  ----------------------
    0                0                       0
    4               33                       1
    9           409113                       2
   14      93928268313                       3
   19  ...485935180313                       4
   24  ...567844940313                       6
   29  ...395300940313                       7
   34  ...323620940313                       8
   39  ...232420940313                       9
   44  ...080420940313                      10
   49  ...920420940313                      12
  ...
   oo  ...920420940313
(End)

Cf. A007489, A003422, A007845, A045748.

Take[Reverse[IntegerDigits[Sum[n!, {n, 1, 500}]]], 100] (* generates first 100 terms *)

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

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A064733 Final digits of A005165(2n) for large n, read from right.

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A065356 Final digits of A065355(n) (in reverse order) for sufficiently large n.

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A364128 Decimal expansion of a constant related to A053529 and A179973.

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A064734 Final digits of A005165(2n+1) for large n, read from right.

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A082648 Consider f(m) = Sum_{k=1..m} k! (A007489) when m is very large; a(n) = n-th digit from end.

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