A099020 Euler-Seidel matrix T(k,n) with start sequence A001147, read by antidiagonals.

1, 1, 0, 2, 1, 1, 4, 2, 1, 0, 10, 6, 4, 3, 3, 26, 16, 10, 6, 3, 0, 76, 50, 34, 24, 18, 15, 15, 232, 156, 106, 72, 48, 30, 15, 0, 764, 532, 376, 270, 198, 150, 120, 105, 105, 2620, 1856, 1324, 948, 678, 480, 330, 210, 105, 0, 9496, 6876, 5020, 3696, 2748, 2070, 1590, 1260, 1050, 945, 945

Offset: 0

Author

Ralf Stephan, Sep 23 2004

Keywords

• nonn
• tabl

Comments

In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences, with the first column the inverse binomial transform of the start sequence.

Examples

			1,   0,  1,  0,   3,   0,   15, ...
1,   1,  1,  3,   3,  15,   15, ...
2,   2,  4,  6,  18,  30,  120, ...
4,   6, 10, 24,  48, 150,  330, ...
10, 16, 34, 72, 198, 480, 1590, ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Crossrefs

First column is A000085, 2nd A013989, main diagonal is in A099021.

Programs

Maple

T:= proc(k, n) option remember; `if`(k=0, `if`(irem(n, 2)=0,
      doublefactorial(n-1), 0), T(k-1, n) +T(k-1, n+1))
    end:
seq(seq(T(d-n, n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 14 2012

Mathematica

t[0, n_?EvenQ] := (n-1)!!; t[0, n_?OddQ] := 0; t[k_, n_] := t[k, n] = t[k-1, n] + t[k-1, n+1]; Table[t[k-n, n], {k, 0, 10}, {n, 0, k}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)

Formula

Recurrence: T(0, 2n) = (2n-1)!!, T(0, 2n+1) = 0, T(k, n) = T(k-1, n) + T(k-1, n+1).