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].
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
Examples
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,
Links
- G. C. Greubel, Rows n=0..100 of triangle, flattened
Crossrefs
Programs
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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
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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
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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 *) -
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
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]
Comments