A039763 Triangle of D-analogs of Stirling numbers of first kind, rows reversed.
1, 1, 0, 1, -2, 1, 1, -6, 11, -6, 1, -12, 50, -84, 45, 1, -20, 150, -520, 809, -420, 1, -30, 355, -2100, 6439, -9390, 4725, 1, -42, 721, -6510, 33019, -92358, 127539, -62370, 1, -56, 1316, -16856, 127694, -578984, 1505524, -1984584, 945945, 1, -72, 2220, -38304, 405174, -2702448, 11228300, -27491616, 34812945, -16216200
Offset: 0
Examples
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins: 1; 1, 0; 1, -2, 1; 1, -6, 11, -6; 1, -12, 50, -84, 45; 1, -20, 150, -520, 809, -420; ...
Links
- Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
Crossrefs
Cf. A039762 (transposed triangle).
Programs
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PARI
row(n) = if(n==0, [1], Vec(prod(i=1, n-1, x-2*i+1)*(x-n+1))); \\ Petros Hadjicostas, Jul 12 2020
Formula
From Petros Hadjicostas, Jul 11 2020: (Start)
T(n,k) = A039762(n,n-k) for k = 0..n.
T(n,0) = 1 for n >= 0.
T(n,n) = (-1)^n*(n-1)*(2*n-3)!! for n >= 2.
T(n,k) = [x^(n-k)] (x - (n - 1)) * Product_{k=1..n-1} (x - (2*k - 1)) for n >= 1 and k = 0..n. (End)
Extensions
More terms from Petros Hadjicostas, Jul 12 2020
Comments