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A233449
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a(n) = Sum_{k=0..n} k! * 2^(n-k).
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4
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1, 3, 8, 22, 68, 256, 1232, 7504, 55328, 473536, 4575872, 49068544, 577138688, 7381298176, 101940887552, 1511556143104, 23945902174208, 403579232444416, 7209532170616832, 136064164750065664, 2705030337676771328, 56501002847062982656, 1237002733471733645312
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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)
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
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EXAMPLE
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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
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MAPLE
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# 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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = sum(k=0, n, k! * 2^(n-k))}; /* Michael Somos, Nov 21 2018 */
(Python) # 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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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