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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].

%I A061312 #15 Sep 08 2022 08:45:03
%S A061312 0,1,1,4,3,2,18,14,11,9,96,78,64,53,44,600,504,426,362,309,265,4320,
%T A061312 3720,3216,2790,2428,2119,1854,35280,30960,27240,24024,21234,18806,
%U A061312 16687,14833,322560,287280,256320,229080,205056,183822,165016,148329
%N 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].
%C A061312 Appears in the (n,k)-matching problem A076731. [_Johannes W. Meijer_, Jul 27 2011]
%H A061312 G. C. Greubel, <a href="/A061312/b061312.txt">Rows n=0..100 of triangle, flattened</a>
%F A061312 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 A061312 T(n,m) = sum(((-1)^j)*binomial(m+1,j)*(n+1-j)!, j=0..m+1) [_Johannes W. Meijer_, Jul 27 2011]
%e A061312 0,
%e A061312 1, 1,
%e A061312 4, 3, 2,
%e A061312 18, 14, 11, 9,
%e A061312 96, 78, 64, 53, 44,
%e A061312 600, 504, 426, 362, 309, 265,
%e A061312 4320, 3720, 3216, 2790, 2428, 2119, 1854,
%e A061312 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833,
%p A061312 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 A061312 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 A061312 (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 A061312 (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 A061312 Cf. A061018.
%Y A061312 From _Johannes W. Meijer_, Jul 27 2011: (Start)
%Y A061312 Columns: A001563, A001564, A001565, A001688, A001689, A023044, A023045, A023046, A023047; A000166, A000255, A055790;
%Y A061312 The row sums equal A193465. (End)
%K A061312 nonn,tabl,easy
%O A061312 0,4
%A A061312 _Wouter Meeussen_, Jun 06 2001