A000023 -id:A000023 - cp's OEIS frontend

A052124 Expansion of e.g.f. exp(-2*x)/(1-x)^3.

1, 1, 4, 16, 88, 568, 4288, 36832, 354688, 3781504, 44199424, 561823744, 7714272256, 113769309184, 1793341407232, 30085661765632, 535170830467072, 10060645294440448, 199287423535808512, 4148644277780217856, 90545807649965080576, 2067407731760475406336, 49285894020028992323584

Offset: 0

N. J. A. Sloane, Jan 23 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64(b).

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

Cf. A052127, A082031.
Cf. A000023, A087981.
Cf. A000153, A137775.

A052124 := proc(n) option remember; if n <=1 then 1 else n*A052124(n-1)+2*(n-1)*A052124(n-2); fi; end; # Detlef Pauly

Table[(n+5)*(n+2)*n!*Sum[(-1)^k*2^(k+2)*(k+3)/(k+5)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)
With[{nn=20},CoefficientList[Series[Exp[(-2x)]/(1-x)^3,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 23 2017 *)

my(x='x+O('x^25)); Vec(serlaplace( exp(-2*x)/(1-x)^3)) \\ Michel Marcus, Oct 25 2021

from math import factorial
from fractions import Fraction
def A052124(n): return int((n+5)*(n+2)*factorial(n)*sum(Fraction((-1 if k&1 else 1)*(k+3)<Chai Wah Wu, Apr 20 2023

a(n) = n*a(n-1) + 2*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003
a(n) = (n+5)*(n+2)*n! * Sum_{k=0..n} (-1)^k*2^(k+2)*(k+3)/(k+5)!. - Vaclav Kotesovec, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+3)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(n) ~ n!*(n+5)*(n+2)/(2*exp(2)). - Vaclav Kotesovec, Jun 15 2013
From Peter Bala, Sep 20 2013: (Start)
a(n) ~ (1/2)*n^2*n!/e^2 for large n.
Letting n -> infinity in the above series for a(n) given by Kotesovec gives the series expansion 1/e^2 = Sum_{k >= 0} (-1)^k*(k+3)*2^(k+3)/(k+5)!.
The sequence b(n) := (1/2)*n!*(n+2)*(n+5) satisfies the recurrence for a(n) given above by Pauly but with the starting values b(0) = 5 and b(1) = 9. This leads to the finite continued fraction expansion a(n) = (1/2)*n!*(n+2)*(n+5)( 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/n)))) ), valid for n >= 2. Letting n -> infinity in the previous result gives the infinite continued fraction expansion 1/e^2 = 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/(n + ...))))). Cf. A082031. (End)
a(n) = A087981(n+2)/(2*(n+1)). - Seiichi Manyama, Apr 25 2025

A263823 a(n) = n!*Sum_{k=0..n} Fibonacci(k-1)/k!, where Fibonacci(-1) = 1, Fibonacci(n) = A000045(n) for n>=0.

Original entry on oeis.org

1, 1, 3, 10, 42, 213, 1283, 8989, 71925, 647346, 6473494, 71208489, 854501957, 11108525585, 155519358423, 2332790376722, 37324646028162, 634518982479741, 11421341684636935, 217005492008104349, 4340109840162091161, 91142306643403921146, 2005130746154886276158

Offset: 0

Vladimir Reshetnikov, Oct 27 2015

nonn

			For n = 3, a(3) = 3!*(Fibonacci(-1)/0! + Fibonacci(0)/1! + Fibonacci(1)/2! + Fibonacci(2)/3!) = 6*(1 + 0 + 1/2 + 1/6) = 10.
For n = 5, Gamma(5+1, phi)*exp(phi) = 120*sqrt(5) + 333 = 240*phi + 213, so a(5) = 213.
G.f. = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 213*x^5 + 1283*x^6 + 8989*x^7 + 71925*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..445
Eric Weisstein's MathWorld, Incomplete Gamma Function.
Eric Weisstein's MathWorld, Fibonacci Number.
Eric Weisstein's MathWorld, Golden Ratio.

Cf. A000045, A111139.

Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

Table[n! Sum[Fibonacci[k-1]/k!, {k, 0, n}], {n, 0, 22}]
Round@Table[(E^(1-GoldenRatio) GoldenRatio Gamma[n+1, 1-GoldenRatio] + E^GoldenRatio Gamma[n+1, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 22}]

a(n) = (Gamma(n+1, 1-phi)*exp(1-phi)*phi+Gamma(n+1, phi)*exp(phi)/phi)/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function, phi=(1+sqrt(5))/2.
a(n) = (phi^(n-1)*hypergeom([1,-n], [], 1-phi)-(-phi)^(1-n)*hypergeom([1,-n], [], phi))/sqrt(5).
Gamma(n+1, phi)*exp(phi) = A111139(n)*phi + a(n).
E.g.f.: (exp(phi*x)/phi+exp(-x/phi)*phi)/(sqrt(5)*(1-x)) = exp(x/2)*(cosh(x*sqrt(5)/2)-sinh(x*sqrt(5)/2)/sqrt(5))/(1-x).
Recurrence: a(0) = 1, a(1) = 1, a(2) = 3, a(n) = (n+1)*a(n-1)+(2-n)*a(n-2)+(2-n)*a(n-3).
a(n) ~ 2*exp(phi-n)*n^(n+1/2)*(1+exp(-sqrt(5))*phi^2)*sqrt(Pi/10)/phi.
0 = a(n)*(+a(n+1) + a(n+2) - 4*a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) + 3*a(n+2) - 5*a(n+3) + a(n+4)) + a(n+2)*(+2*a(n+2) - a(n+4)) + a(n+3)*(+a(n+3)) if n>=0. - Michael Somos, Oct 30 2015

A233449 a(n) = Sum_{k=0..n} k! * 2^(n-k).

Original entry on oeis.org

1, 3, 8, 22, 68, 256, 1232, 7504, 55328, 473536, 4575872, 49068544, 577138688, 7381298176, 101940887552, 1511556143104, 23945902174208, 403579232444416, 7209532170616832, 136064164750065664, 2705030337676771328, 56501002847062982656, 1237002733471733645312

Offset: 0

Vaclav Kotesovec, Dec 10 2013

nonn

Sequence b := A000023 recursion may be extended to n<0 using b(n) = (-1)^m * (x + a(m-1) / 2^m) / m! where m = -1-n and x is arbitrary. - Michael Somos, Nov 21 2018

For n >= 4, we have that (n - 1)*((n - 2)! - 2^(n - 3) + a(n - 2)) corresponds to the number of random walk labelings of the wheel graph W_n (i.e., the graph formed by connecting a vertex to all the vertices of the cycle graph C_(n-1)). - Sela Fried, Apr 18 2023

			G.f. = 1 + 3*x + 8*x^2 + 22*x^3 + 68*x^4 + 256*x^5 + 1232*x^6 + ... - _Michael Somos_, Nov 21 2018

Seiichi Manyama, Table of n, a(n) for n = 0..449
Sela Fried and Toufik Mansour, Further results on random walk labelings, arXiv:2305.09971 [math.CO], 2023.

Cf. A000023, A004400, A003422, A123642.

# Assuming sequence starts with a(0) = 0 (see comment in formula section).
egf := exp(2*x - 2)*(Ei(1, 2*x - 2) - Ei(1, -2)):
assume(x > 0): ser := series(egf, x, 24):
seq(n! * simplify(coeff(ser, x, n)), n = 0..23);  # Peter Luschny, Mar 05 2024

Table[Sum[k!*2^(n-k), {k, 0, n}], {n, 0, 20}]
Table[FullSimplify[(2^(1+n)*(I*Pi + ExpIntegralEi[2]) + ExpIntegralE[2+n,-2]*Gamma[2+n])/E^2], {n, 0, 20}]
max = 20; Clear[g]; g[max+2] = 1; g[k_] := g[k] = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/g[k+1] ))); gf = g[0]/(1-2*x); CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jan 13 2015, after Sergei N. Gladkovskii *)

{a(n) = sum(k=0, n, k! * 2^(n-k))}; /* Michael Somos, Nov 21 2018 */

# Assuming sequence starts with a(0) = 0 (see formula section).
def a(n):
    if n == 0: return 0
    s = f = 1
    for k in range(1, n):
        f *= k / 2
        s += f
    return round(2**(n - 1) * s)  # 'round' only to assure the integer type.
print([a(n) for n in range(24)])  # Peter Luschny, Mar 05 2024

a(n) = (n+2)*a(n-1) - 2*n*a(n-2).
a(n) ~ n!.
G.f.: G(0)/(1-2*x), where G(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/G(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Jan 13 2015
0 = a(n)*(+4*a(n+1) -6*a(n+2) +2*a(n+3)) +a(n+1)*(+2*a(n+1) -a(n+2) -a(n+3)) + a(n+2)*(+a(n+2)) if n>=0. - Michael Somos, Nov 21 2018
E.g.f.: 2*e^(2*x-2)*(Integral_{t=0..1} (e^(2t) - e^(2*(1-x)*t))/t dt - log(1-x)) + 1/(1-x). - Sela Fried, May 23 2023 (Theorem 14 in the reference)
a(0) = 1; a(n) = 2*a(n-1) + n!. - Seiichi Manyama, Dec 29 2023
We assume that '0' is prepended to a, i.e., a = 0, 1, 3, 8, ... with offset 0. The terms are the coefficients of the exponential generating function 2*exp(2*x - 2)*(Ei(m, 2*x - 2) - Ei(m, -2)) where Ei is the exponential integral and m = 1. If m = 0 then, up to sign, the sequence A123642 is generated. - Peter Luschny, Mar 05 2024

A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

Original entry on oeis.org

1, 3, 14, 82, 576, 4688, 43264, 445632, 5062016, 62812288, 844863744, 12239474432, 189939644416, 3142842052608, 55223903596544, 1026805938614272, 20139224002953216, 415503046091767808, 8994794537935765504, 203848794955954716672, 4826475681472562855936, 119162892472107134353408

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A001909, A383382, A383383, A383384.

Cf. A000023, A052124, A087981, A383380.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^5))

a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = 1, a(1) = 3; a(n) = (n+2)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A292977 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 6, 1, -2, 2, 2, 24, 1, -3, 5, -2, 9, 120, 1, -4, 10, -12, 8, 44, 720, 1, -5, 17, -34, 33, 8, 265, 5040, 1, -6, 26, -74, 120, -78, 112, 1854, 40320, 1, -7, 37, -138, 329, -424, 261, 656, 14833, 362880, 1, -8, 50, -232, 744, -1480, 1552, -360, 5504, 133496, 3628800

Offset: 0

Ilya Gutkovskiy, Sep 27 2017

A(n,k) is the k-th inverse binomial transform of A000142 evaluated at n.

Can be considered as extension of the array A089258 to columns with negative indices via A089258(n,k) = A(n,-k) or, vice versa, A(n,k) = A089258(n,-k). - Max Alekseyev, Mar 06 2018

			Square array begins:
n=0:    1,   1,   1,    1,     1,      1,  ...
n=1:    1,   0,  -1,   -2,    -3,     -4,  ...
n=2:    2,   1,   2,    5,    10,     17,  ...
n=3:    6,   2,  -2,  -12,   -34,    -74,  ...
n=4:   24,   9,   8,   33,   120,    329,  ...
n=5:  120,  44,   8,  -78,  -424,  -1480,  ...
...
E.g.f. of column k: A_k(x) = 1 + (1 - k)*x/1! +  (k^2 - 2*k + 2)*x^2/2! + (-k^3 + 3*k^2 - 6*k + 6) x^3/3! + (k^4 - 4*k^3 + 12*k^2 - 24*k + 24)*x^4/4! + ...

N. J. A. Sloane, Transforms

Columns: A000142 (k=0), A000166 (k=1), A000023 (k=2), A010843 (k=3, with offset 0).
Main diagonal: A134095 (absolute values).
Cf. A080955, A089258.

Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
FullSimplify[Table[Function[k, Exp[-k] Gamma[n + 1, -k]][j - n], {j, 0, 10}, {n, 0, j}]] // Flatten

T(n, k) = n! * Sum_{j=0..n} (-k)^j/j!. - Max Alekseyev, Mar 06 2018

E.g.f. of column k: exp(-k*x)/(1 - x).

A335391 Square array read by antidiagonals downwards: for n >= 2, T(k,n) is the number of permutations of [k+n] that differ in every position from both the identity permutation and a permutation consisting of k 1-cycles and one n-cycle.

Original entry on oeis.org

2, -1, 0, 0, 1, 2, 1, 0, 1, 4, 2, 3, 4, 7, 18, 13, 16, 19, 24, 35, 88, 80, 95, 114, 137, 168, 221, 530, 579, 672, 783, 916, 1077, 1280, 1589, 3708, 4738, 5397, 6164, 7061, 8114, 9359, 10860, 12979, 29666, 43387, 48704, 54773, 61720, 69697, 78888, 89527, 101976, 118663, 266992

Offset: 0

William P. Orrick, Jun 04 2020

The number of permutations of [k+n] that differ in every position from both the identity permutation and a permutation consisting of k 1-cycles and s cycles of lengths p_1, p_2, ... p_s, with p_j >= 2 and p_1+p_2+...+p_s = n, can be expressed as Sum T(k,p_1+-p_2+-...+-p_s), where the sum is over all 2^(s-1) choices of sign and where T(k,-n) = T(k,n) (Touchard).
The first of Touchard's formulas for T(k,n) involves A034807, the number of k-matchings of C_n (A213234 or A127677 with sign included) and A047920, the k-th differences of the factorial numbers.
A slightly different formula, due to Wyman and Moser in the k=0 case, involves A213234 and A000023.
The first column is twice A000166 (twice the number of derangements of [k]); the second column is A105926 (first differences of A000166); the third column is A331007 (with offset 2); the first row is A102761 (the ménage numbers); the second row is A000270.

			Array starts:
k/n |    0     1      2      3       4         5          6           7
-----------------------------------------------------------------------
0   |    2    -1      0      1       2        13         80         579
1   |    0     1      0      3      16        95        672        5397
2   |    2     1      4     19     114       783       6164       54773
3   |    4     7     24    137     916      7061      61720      602955
4   |   18    35    168   1077    8114     69697     671736     7172007
5   |   88   221   1280   9359   78888    749547    7913440    91815601
6   |  530  1589  10860  89527  837794   8741875  100478588  1260186153
7   | 3708 12979 101976 938181 9669196 110058257 1369406616 18475560567
There are T(1,3)=3 permutations that differ from 1234=(1)(2)(3)(4) and 1342=(1)(234) in every position: 2413, 3421, and 4123.

Mathematics Stack Exchange, Double derangement and the other unfamous kind of derangement.
J. Touchard, Sur un problème de permutations, Comptes Rendus Acad. Sci. Paris, 198 (1934) 631-633.
Max Wyman and Leo Moser, On the problème des ménages, Canadian Journal of Mathematics, 10 (1958) 468-480.

Cf. A034807, A213234, A127677, A047920, A000023, A000166, A105926, A331007, A102761, A000270.

T := proc(n,k) local t; t := proc(n, k) option remember;
   simplify((n + k)!*hypergeom([-n], [-n - k], -1)) end:
   if k = 0 then return 2*t(n, 0) fi;
   add((-1)^j*(2*k)/(2*k-j)*binomial(2*k-j, j)*t(n, k-j), j=0 ..k) end:
seq(lprint(seq(T(n, k), k=0..7)),n=0..7); # Peter Luschny, Jul 22 2020

f(k, n) = sum(j=0, k, (-1)^j*binomial(k, j)*(n+k-j)!);
T(k, n) = if (n==0, 2*f(k, 0), sum(j=0, n, (-1)^j*(2*n)/(2*n-j)*binomial(2*n-j, j)*f(k, n-j)));
matrix(7, 7,n, k, T(n-1,k-1))
\\ Michel Marcus, Jun 26 2020

def f(k,n):
    return sum((-1)^j*binomial(k,j)*factorial(n+k-j) for j in range(0,k+1))
def T(k,n):
    if n==0:
        return 2*f(k,0)
    else:
        return sum((-1)^j*(2*n)/(2*n-j)*binomial(2*n-j,j)*f(k,n-j) for j in range(0,n+1))

T(k,0) = 2*nu(k,k), T(k,n>0) = Sum_{j=0..n} A213234(2*n,j)*nu(k,k+n-j) = Sum_{j=0..n} (-1)^j*2*n/(2*n-j)*binomial(2*n-j,j)*nu(k,k+n-j) where nu(k,k+n) = A047920(k+n,k) = Sum_{j=0..k} (-1)^j*binomial(k,j)*(k+n-j)! (Touchard).
T(k,n) = 2*cos(2*n * arccos(1/2*sqrt(x))) = 2*Chebyshev_T(2*n,sqrt(x)/2), where, after expanding in powers of x, x^m gets replaced by nu(k,k+m) (Touchard).
T(k,n) = 2*(-1)^n*Sum_{j=0..n} (-1)^j*(Product_{r=0..j} n^2-r^2)/(2*j)!*nu(k,k+j) (Touchard).
T(k,n) = 2*Integral_{x=0..oo} e^(-x^2) * (x^2-1)^k * x * ((x+sqrt(x^2-4))^(2*n)+(x-sqrt(x^2-4))^(2*n)) / 2^(2*n) dx (Touchard).
T(k,0) = 2*Sum_{j=0..h} binomial(h,j)*k(j), T(k,n) = Sum_{i>=0} A213234(n,i)*Sum_{j=0..h} binomial(h,j)*k(n-2*i+j) = Sum_{i>=0} (-1)^i*n/(n-i)*binomial(n-i,i)*Sum_{j=0..h} binomial(h,j)*k(n-2*i+j) where k(n) = A000023(n) = n! * Sum_{i=0..n} (-2)^i / i! (k=0 case due to Wyman and Moser)
T(k+1,n+1) = T(k,n)+T(k,n+1)+T(k,n+2): This holds for all integers n if one defines T(k,-n) = T(k,n).
T(k,0) = 2*A000166(k).
T(k,1) = A105926(k).
T(k,2) = A331007(k+2).
T(0,n) = A102761(n).
T(1,n) = A000270(n).

A346397 Expansion of e.g.f. -log(1 - x) * exp(-2*x).

Original entry on oeis.org

0, 1, -3, 8, -18, 44, -80, 272, 112, 5280, 38464, 414336, 4573184, 55680000, 731374592, 10335551488, 156303374336, 2518984953856, 43099088904192, 780268881068032, 14902336355991552, 299452809651617792, 6315501510330286080, 139485953831281098752, 3219718099932087844864

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A000023, A002741, A335111, A346398.

nmax = 24; CoefficientList[Series[-Log[1 - x] Exp[-2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-2)^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 24}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i-3)*v[i]+2*(i-1)*v[i-1]+(-2)^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} (-2)^k / ((n-k) * k!).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002741(k).
a(0) = 0, a(1) = 1, a(n) = (n-3) * a(n-1) + 2 * (n-1) * a(n-2) + (-2)^(n-1). - Seiichi Manyama, May 27 2022
a(n) ~ exp(-2) * (n-1)!. - Vaclav Kotesovec, Jun 08 2022

A033166 Incrementally largest terms in the continued fraction for zeta(3).

Original entry on oeis.org

1, 4, 18, 30, 428, 458, 527, 542, 659, 922, 1355, 1817, 3292, 4635, 6841, 162050, 197106, 270086, 270919, 344064, 2079442, 5492021, 196362299, 220581908, 1292221854, 5841289122, 11886461834

Offset: 1

N. J. A. Sloane

Eric Weisstein's World of Mathematics, Apery's Constant Continued Fraction

Cf. A013631 (continued fraction of zeta(3)).
Cf. A229055 (positions of incrementally largest terms in the c.f. of zeta(3)).
Cf. A033165, A000023, A229055.

More terms from Eric W. Weisstein, Aug 23 2000
More terms from Robert Gerbicz, Aug 22 2006
More terms from Eric W. Weisstein, Sep 16 2013
a(27) = 11886461834 from Eric W. Weisstein, Sep 17 2013

A111139 a(n) = n!*Sum_{k=0..n} Fibonacci(k)/k!.

Original entry on oeis.org

0, 1, 3, 11, 47, 240, 1448, 10149, 81213, 730951, 7309565, 80405304, 964863792, 12543229529, 175605213783, 2634078207355, 42145251318667, 716469272418936, 12896446903543432, 245032491167329389, 4900649823346594545

Offset: 0

Vladeta Jovovic, Oct 17 2005

Eigensequence of a triangle with the Fibonacci series as the left border, the natural numbers (1, 2, 3, ...) as the right border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's MathWorld, Incomplete Gamma Function.
Eric Weisstein's MathWorld, Fibonacci Number.
Eric Weisstein's MathWorld, Golden Ratio.

Cf. A000045, A263823.

Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

a:=n->sum(fibonacci (j)*n!/j!,j=0..n):seq(a(n),n=0..20); # Zerinvary Lajos, Mar 19 2007

f[n_] := n!*Sum[Fibonacci[k]/k!, {k, 0, n}]; Table[ f[n], {n, 0, 20}] (* or *)
Simplify[ Range[0, 20]!CoefficientList[ Series[2/Sqrt[5]*Exp[x/2]*Sinh[Sqrt[5]*x/2]/(1 - x), {x, 0, 20}], x]] (* Robert G. Wilson v, Oct 21 2005 *)
Module[{nn=20,fibs,fct},fct=Range[0,nn]!;fibs=Accumulate[ Fibonacci[ Range[ 0,nn]]/fct];Times@@@Thread[{fct,fibs}]] (* Harvey P. Dale, Feb 19 2014 *)
Round@Table[(E^GoldenRatio Gamma[n+1, GoldenRatio] - E^(1-GoldenRatio) Gamma[n+1, 1-GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 27 2015 *)

vector(100, n, n--; n!*sum(k=0, n, fibonacci(k)/k!)) \\ Altug Alkan, Oct 28 2015

E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2)/(1-x).
Recurrence: a(n) = (n+1)*a(n-1) - (n-2)*a(n-2) - (n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2*sqrt(e/5)*sinh(sqrt(5)/2)*n!. - Vaclav Kotesovec, Oct 18 2012
From Vladimir Reshetnikov, Oct 27 2015: (Start)
Let phi=(1+sqrt(5))/2.
a(n) = (phi^n*hypergeom([1,-n], [], 1-phi)-(1-phi)^n*hypergeom([1,-n], [], phi))/sqrt(5).
a(n) = (exp(phi)*Gamma(n+1, phi)-exp^(1-phi)*Gamma(n+1, 1-phi))/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function.
Gamma(n+1, phi)*exp(phi) = a(n)*phi + A263823(n).
a(n) ~ exp(phi-n)*n^(n+1/2)*sqrt(2*Pi/5)*(1-exp(-sqrt(5))).
(End)

A277431 Expansion of e.g.f.: cosh(sqrt(2)*x)/(1-x).

Original entry on oeis.org

1, 1, 4, 12, 52, 260, 1568, 10976, 87824, 790416, 7904192, 86946112, 1043353408, 13563594304, 189890320384, 2848354805760, 45573676892416, 774752507171072, 13945545129079808, 264965357452516352, 5299307149050328064, 111285450130056889344, 2448279902861251567616

Offset: 0

Vladimir Reshetnikov, Oct 14 2016

nonn

			G.f. = 1 + x + 4*x^2 + 12*x^3 + 52*x^4 + 260*x^5 + 1568*x^6 + ... - _Michael Somos_, Oct 01 2018

G. C. Greubel, Table of n, a(n) for n = 0..448
Eric Weisstein's MathWorld, Incomplete Gamma Function

Cf. A000023, A263823, A277345, A277432.

I:=[1,4,12]; [1] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 30 2018

Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] + Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/2, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Table[SeriesCoefficient[Cosh[Sqrt[2] x]/(1 - x), {x, 0, n}] n!, {n, 0, 20}]
a[ n_] := If[ n < 0, 0, n! Sum[ 2^k / (2 k)!, {k, 0, n/2}]]; (* Michael Somos, Oct 01 2018 *)
With[{nn=30},CoefficientList[Series[Cosh[x Sqrt[2]]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 13 2024 *)

x='x+O('x^30); Vec(serlaplace(cosh(sqrt(2)*x)/(1-x))) \\ G. C. Greubel, Sep 30 2018

{a(n) = if( n<0, 0, n! * sum(k=0, n\2, 2^k / (2*k)!))}; /* Michael Somos, Oct 01 2018 */

a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) + Gamma(n+1, -sqrt(2))/exp(sqrt(2)))/2.
a(n) ~ sqrt(2*Pi)*cosh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = a(n) + sqrt(2)*A277432(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = a(n) - sqrt(2)*A277432(n).
a(2*n+1) = (2*n+1)*a(2*n).
0 = a(n)*(+4*a(n+1) -4*a(n+2) -6*a(n+3) +2*a(n+4)) +a(n+1)*(+4*a(n+1) +2*a(n+2) -4*a(n+3)) +a(n+2)*(+2*a(n+2) +a(n+3) -a(n+4)) + a(n+3)*(+a(n+3)) for all n>-3. - Michael Somos, Oct 01 2018

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A052124 Expansion of e.g.f. exp(-2*x)/(1-x)^3.

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A233449 a(n) = Sum_{k=0..n} k! * 2^(n-k).

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A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

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A292977 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x).

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A335391 Square array read by antidiagonals downwards: for n >= 2, T(k,n) is the number of permutations of [k+n] that differ in every position from both the identity permutation and a permutation consisting of k 1-cycles and one n-cycle.

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A346397 Expansion of e.g.f. -log(1 - x) * exp(-2*x).

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