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 A306150 Row sums of A306015. 2
 0, 2, 4, 14, 56, 282, 1692, 11846, 94768, 852914, 8529140, 93820542, 1125846504, 14636004554, 204904063756, 3073560956342, 49176975301472, 836008580125026, 15048154442250468, 285914934402758894, 5718298688055177880, 120084272449158735482, 2641853993881492180604 (list; graph; refs; listen; history; text; internal format)
 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:=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

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Last modified July 25 14:22 EDT 2024. Contains 374609 sequences. (Running on oeis4.)