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A061312
Triangle T[n,m]: T[n,-1] = 0; T[0,0] = 0; T[n,0] = n*n!; T[n,m] = T[n,m-1] - T[n-1,m-1].
5
0, 1, 1, 4, 3, 2, 18, 14, 11, 9, 96, 78, 64, 53, 44, 600, 504, 426, 362, 309, 265, 4320, 3720, 3216, 2790, 2428, 2119, 1854, 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833, 322560, 287280, 256320, 229080, 205056, 183822, 165016, 148329
OFFSET
0,4
COMMENTS
Appears in the (n,k)-matching problem A076731. [Johannes W. Meijer, Jul 27 2011]
FORMULA
T[n,m] = T[n,m-1]-T[n-1,m-1] with T[n,-1] = 0 and T[n,0] = A001563(n) = n*n!
T(n,m) = sum(((-1)^j)*binomial(m+1,j)*(n+1-j)!, j=0..m+1) [Johannes W. Meijer, Jul 27 2011]
EXAMPLE
0,
1, 1,
4, 3, 2,
18, 14, 11, 9,
96, 78, 64, 53, 44,
600, 504, 426, 362, 309, 265,
4320, 3720, 3216, 2790, 2428, 2119, 1854,
35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833,
MAPLE
A061312 := proc(n, m): add(((-1)^j)*binomial(m+1, j)*(n+1-j)!, j=0..m+1) end: seq(seq(A061312(n, m), m=0..n), n=0..7); # Johannes W. Meijer, Jul 27 2011
MATHEMATICA
T[n_, k_]:= Sum[(-1)^j*Binomial[k + 1, j]*(n + 1 - j)!, {j, 0, k + 1}]; Table[T[n, k], {n, 0, 100}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 13 2018 *)
PROG
(PARI) for(n=0, 20, for(k=0, n, print1(sum(j=0, k+1, (-1)^j*binomial(k+1, j) *(n-j+1)!), ", "))) \\ G. C. Greubel, Aug 13 2018
(Magma) [[(&+[(-1)^j*Binomial(k+1, j)*Factorial(n-j+1): j in [0..k+1]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Aug 13 2018
CROSSREFS
Cf. A061018.
From Johannes W. Meijer, Jul 27 2011: (Start)
The row sums equal A193465. (End)
Sequence in context: A239020 A293211 A330778 * A019130 A245348 A174551
KEYWORD
nonn,tabl,easy
AUTHOR
Wouter Meeussen, Jun 06 2001
STATUS
approved