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%I #23 Dec 19 2021 07:36:35
%S 0,1,1,1,2,1,4,6,3,1,15,24,12,4,1,76,120,60,20,5,1,455,720,360,120,30,
%T 6,1,3186,5040,2520,840,210,42,7,1,25487,40320,20160,6720,1680,336,56,
%U 8,1,229384,362880,181440,60480,15120,3024,504,72,9,1
%N Exponential series expansion of (exp(x*y) + sinh(x) - cosh(x))/(1 - x).
%C From _David Callan_, Dec 18 2021: (Start)
%C For 0 <= k <= n, T(n,k) is the number of nonderangements of size n in which k of the fixed points are colored red. In particular, with D_n the derangement number A000166(n), T(n,0) = n! - D_n. For a general example, T(3,1) = 6 counts the colored permutations R23, R32, 1R3, 3R1, 12R, 21R where the red fixed points are indicated by "R".
%C For n >= k >= 1, T(n,k) = n!/k!. Proof. In a colored permutation, such as 3R7R516 counted by T(n,k) with n = 7 and k = 2, the R's indicate (red) fixed points and so no information is lost by rank ordering the remaining entries while retaining the placement of the R's: 2R5R314. The result is a permutation of the set consisting of 1,2,...,n-k and k R's; there are n!/k! such permutations and the process is reversible. QED. (End)
%H G. C. Greubel, <a href="/A306015/b306015.txt">Rows n=0..99 of triangle, flattened</a>
%e n | k = 0 1 2 3 4 5 6 7 8 9
%e --+----------------------------------------------------------
%e 0 | 0
%e 1 | 1, 1
%e 2 | 1, 2, 1
%e 3 | 4, 6, 3, 1
%e 4 | 15, 24, 12, 4, 1
%e 5 | 76, 120, 60, 20, 5, 1
%e 6 | 455, 720, 360, 120, 30, 6, 1
%e 7 | 3186, 5040, 2520, 840, 210, 42, 7, 1
%e 8 | 25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1
%e 9 | 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
%p gf := (exp(x*y) + sinh(x) - cosh(x))/(1 - x):
%p ser := series(gf, x, 16): L := [seq(n!*coeff(ser, x, n), n=0..14)]:
%p seq(seq(coeff(L[k+1], y, n), n=0..k), k=0..12);
%t Join[{0}, With[{nmax = 15}, CoefficientList[CoefficientList[Series[ (Exp[x*y] + Sinh[x] - Cosh[x])/(1 - x), {x, 0, nmax}, {y, 0, nmax}], x], y ]*Range[0, nmax]!] // Flatten ] (* _G. C. Greubel_, Jul 18 2018 *)
%Y A094587 with an extra first column A002467.
%Y Row sums are A306150.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jun 23 2018
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