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A306015 Exponential series expansion of (exp(x*y) + sinh(x) - cosh(x))/(1 - x).

%I A306015 #23 Dec 19 2021 07:36:35
%S A306015 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 A306015 6,1,3186,5040,2520,840,210,42,7,1,25487,40320,20160,6720,1680,336,56,
%U A306015 8,1,229384,362880,181440,60480,15120,3024,504,72,9,1
%N A306015 Exponential series expansion of (exp(x*y) + sinh(x) - cosh(x))/(1 - x).
%C A306015 From _David Callan_, Dec 18 2021: (Start)
%C A306015 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 A306015 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 A306015 G. C. Greubel, <a href="/A306015/b306015.txt">Rows n=0..99 of triangle, flattened</a>
%e A306015   n |  k = 0       1       2      3      4     5    6   7  8  9
%e A306015   --+----------------------------------------------------------
%e A306015   0 |      0
%e A306015   1 |      1,      1
%e A306015   2 |      1,      2,      1
%e A306015   3 |      4,      6,      3,     1
%e A306015   4 |     15,     24,     12,     4,     1
%e A306015   5 |     76,    120,     60,    20,     5,    1
%e A306015   6 |    455,    720,    360,   120,    30,    6,   1
%e A306015   7 |   3186,   5040,   2520,   840,   210,   42,   7,  1
%e A306015   8 |  25487,  40320,  20160,  6720,  1680,  336,  56,  8, 1
%e A306015   9 | 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
%p A306015 gf := (exp(x*y) + sinh(x) - cosh(x))/(1 - x):
%p A306015 ser := series(gf, x, 16): L := [seq(n!*coeff(ser, x, n), n=0..14)]:
%p A306015 seq(seq(coeff(L[k+1], y, n), n=0..k), k=0..12);
%t A306015 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 A306015 A094587 with an extra first column A002467.
%Y A306015 Row sums are A306150.
%K A306015 nonn,tabl
%O A306015 0,5
%A A306015 _Peter Luschny_, Jun 23 2018