%I #15 Sep 08 2022 08:45:03

%S 0,1,1,4,3,2,18,14,11,9,96,78,64,53,44,600,504,426,362,309,265,4320,

%T 3720,3216,2790,2428,2119,1854,35280,30960,27240,24024,21234,18806,

%U 16687,14833,322560,287280,256320,229080,205056,183822,165016,148329

%N 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].

%C Appears in the (n,k)-matching problem A076731. [_Johannes W. Meijer_, Jul 27 2011]

%H G. C. Greubel, <a href="/A061312/b061312.txt">Rows n=0..100 of triangle, flattened</a>

%F 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!

%F T(n,m) = sum(((-1)^j)*binomial(m+1,j)*(n+1-j)!, j=0..m+1) [_Johannes W. Meijer_, Jul 27 2011]

%e 0,

%e 1, 1,

%e 4, 3, 2,

%e 18, 14, 11, 9,

%e 96, 78, 64, 53, 44,

%e 600, 504, 426, 362, 309, 265,

%e 4320, 3720, 3216, 2790, 2428, 2119, 1854,

%e 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833,

%p 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

%t 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 *)

%o (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

%o (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

%Y Cf. A061018.

%Y From _Johannes W. Meijer_, Jul 27 2011: (Start)

%Y Columns: A001563, A001564, A001565, A001688, A001689, A023044, A023045, A023046, A023047; A000166, A000255, A055790;

%Y The row sums equal A193465. (End)

%K nonn,tabl,easy

%O 0,4

%A _Wouter Meeussen_, Jun 06 2001