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

%I A233449 #37 Mar 05 2024 16:43:28
%S A233449 1,3,8,22,68,256,1232,7504,55328,473536,4575872,49068544,577138688,
%T A233449 7381298176,101940887552,1511556143104,23945902174208,403579232444416,
%U A233449 7209532170616832,136064164750065664,2705030337676771328,56501002847062982656,1237002733471733645312
%N A233449 a(n) = Sum_{k=0..n} k! * 2^(n-k).
%C A233449 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
%C A233449 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
%H A233449 Seiichi Manyama, <a href="/A233449/b233449.txt">Table of n, a(n) for n = 0..449</a>
%H A233449 Sela Fried and Toufik Mansour, <a href="https://arxiv.org/abs/2305.09971">Further results on random walk labelings</a>, arXiv:2305.09971 [math.CO], 2023.
%F A233449 a(n) = (n+2)*a(n-1) - 2*n*a(n-2).
%F A233449 a(n) ~ n!.
%F A233449 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
%F A233449 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
%F A233449 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)
%F A233449 a(0) = 1; a(n) = 2*a(n-1) + n!. - _Seiichi Manyama_, Dec 29 2023
%F A233449 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
%e A233449 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
%p A233449 # Assuming sequence starts with a(0) = 0 (see comment in formula section).
%p A233449 egf := exp(2*x - 2)*(Ei(1, 2*x - 2) - Ei(1, -2)):
%p A233449 assume(x > 0): ser := series(egf, x, 24):
%p A233449 seq(n! * simplify(coeff(ser, x, n)), n = 0..23);  # _Peter Luschny_, Mar 05 2024
%t A233449 Table[Sum[k!*2^(n-k), {k, 0, n}], {n, 0, 20}]
%t A233449 Table[FullSimplify[(2^(1+n)*(I*Pi + ExpIntegralEi[2]) + ExpIntegralE[2+n,-2]*Gamma[2+n])/E^2], {n, 0, 20}]
%t A233449 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_ *)
%o A233449 (PARI) {a(n) = sum(k=0, n, k! * 2^(n-k))}; /* _Michael Somos_, Nov 21 2018 */
%o A233449 (Python) # Assuming sequence starts with a(0) = 0 (see formula section).
%o A233449 def a(n):
%o A233449     if n == 0: return 0
%o A233449     s = f = 1
%o A233449     for k in range(1, n):
%o A233449         f *= k / 2
%o A233449         s += f
%o A233449     return round(2**(n - 1) * s)  # 'round' only to assure the integer type.
%o A233449 print([a(n) for n in range(24)])  # _Peter Luschny_, Mar 05 2024
%Y A233449 Cf. A000023, A004400, A003422, A123642.
%K A233449 nonn
%O A233449 0,2
%A A233449 _Vaclav Kotesovec_, Dec 10 2013