A039762 Triangle of D-analogs of Stirling numbers of first kind.

1, 0, 1, 1, -2, 1, -6, 11, -6, 1, 45, -84, 50, -12, 1, -420, 809, -520, 150, -20, 1, 4725, -9390, 6439, -2100, 355, -30, 1, -62370, 127539, -92358, 33019, -6510, 721, -42, 1, 945945, -1984584, 1505524, -578984, 127694, -16856, 1316, -56, 1, -16216200, 34812945, -27491616, 11228300, -2702448, 405174, -38304, 2220, -72, 1

Offset

0,5

Links

Table of n, a(n) for n=0..54.

Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Formula

From Petros Hadjicostas, Jul 11 2020: (Start)

T(n,k) = [x^k] (x - (n - 1)) * Product_{k=1..n-1} (x - (2*k - 1)) for n >= 1 with T(0,0) = 1. (Empty products equal 1.)

Let R(n,k) = A039757(n,k) = A039758(n,n-k). Then, for n >= 1:

T(n,0) = -(n - 1)*R(n-1,0);

T(n,k) = R(n-1,k-1) - (n - 1)*R(n-1,k) for k = 1..n-1;

T(n,n) = R(n-1, n-1) = 1.

As a result, for n >= 2, T(n,0) = (-1)^n*(n-1)*(2*n-3)!!. (End)

Example

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
     1;
     0,   1;
     1,  -2,    1;
    -6,  11,   -6,   1;
    45, -84,   50, -12,   1;
  -420, 809, -520, 150, -20, 1;
  ...

Prog

(PARI) row(n) = if(n==0, [1], Vecrev(prod(i=1, n-1, x-2*i+1)*(x-n+1))); \\ Petros Hadjicostas, Jul 12 2020

Crossrefs

Cf. A039757, A039758, A039763 (transposed triangle).

Sequence in context: A305512 A121927 A200265 * A039795 A283746 A049949

Adjacent sequences: A039759 A039760 A039761 * A039763 A039764 A039765

Keyword

tabl,sign,easy,nice

Author

Ruedi Suter (suter(AT)math.ethz.ch)

Extensions

More terms from Petros Hadjicostas, Jul 12 2020

Status

approved