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

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, 6, 1, 3186, 5040, 2520, 840, 210, 42, 7, 1, 25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset: 0

Author

Peter Luschny, Jun 23 2018

Keywords

• nonn
• tabl

Comments

From David Callan, Dec 18 2021: (Start)

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".

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)

Examples

			  n |  k = 0       1       2      3      4     5    6   7  8  9
  --+----------------------------------------------------------
  0 |      0
  1 |      1,      1
  2 |      1,      2,      1
  3 |      4,      6,      3,     1
  4 |     15,     24,     12,     4,     1
  5 |     76,    120,     60,    20,     5,    1
  6 |    455,    720,    360,   120,    30,    6,   1
  7 |   3186,   5040,   2520,   840,   210,   42,   7,  1
  8 |  25487,  40320,  20160,  6720,  1680,  336,  56,  8, 1
  9 | 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
		

Links

• G. C. Greubel, Rows n=0..99 of triangle, flattened

Crossrefs

A094587 with an extra first column A002467.

Row sums are A306150.

Programs

Maple

gf := (exp(x*y) + sinh(x) - cosh(x))/(1 - x):
ser := series(gf, x, 16): L := [seq(n!*coeff(ser, x, n), n=0..14)]:
seq(seq(coeff(L[k+1], y, n), n=0..k), k=0..12);

Mathematica

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 *)