A306150 Row sums of A306015.

0, 2, 4, 14, 56, 282, 1692, 11846, 94768, 852914, 8529140, 93820542, 1125846504, 14636004554, 204904063756, 3073560956342, 49176975301472, 836008580125026, 15048154442250468, 285914934402758894, 5718298688055177880, 120084272449158735482, 2641853993881492180604

Offset

0,2

Comments

a(n) is the number of nonderangements of size n in which each fixed point is colored red or blue. For example, with n = 3, the derangements are 231 and 312 and they don't count, the permutations 132, 321, 213 each have 1 fixed point and hence 2 colorings, and the identity 123 with 3 fixed points has 8 colorings, yielding a(3) = 3*2 + 8 = 14 colorings altogether. - David Callan, Dec 19 2021

Links

G. C. Greubel, Table of n, a(n) for n = 0..448

Formula

a(n) = e * Gamma(n + 1, 1) - !(n).

a(n) = Gamma(n + 1, 1) * e - Gamma(n + 1, -1) / e.

a(n) = n*a(n-1) + a(n-2) - (n-2)*a(n-3) for n >= 3.

a(n) = n! [x^n] 2*sinh(x)/(1-x).

a(n) = 2*A186763(n) = (-1)^(n+1)*2*A009628(n) = A000522(n) - A000166(n).

Maple

egf := 2*sinh(x)/(1-x): ser := series(egf, x, 24):
seq(n!*coeff(ser, x, n), n=0..22);

Mathematica

Table[Exp[1] Gamma[n+1, 1] - Subfactorial[n], {n, 0, 22}]
With[{nmax = 50}, CoefficientList[Series[2*Sinh[x]/(1 - x), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 18 2018 *)

Prog

(Sage)
@cached_function
def a(n):
    if n<3: return 2*n
    return n*a(n-1)+a(n-2)-(n-2)*a(n-3)
[a(n) for n in (0..22)]
(PARI) x='x+O('x^30); concat([0], Vec(serlaplace(2*sinh(x)/(1 - x)))) \\ G. C. Greubel, Jul 18 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:= [0] cat Coefficients(R!(2*Sinh(x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 18 2018

Crossrefs

Cf. A000166, A000522, A009628, A186763.

Cf. A306015.

Sequence in context: A367558 A109154 A030853 * A030962 A030830 A059687

Adjacent sequences: A306147 A306148 A306149 * A306151 A306152 A306153

Keyword

nonn

Author

Peter Luschny, Jun 23 2018

Status

approved