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

0,4

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.

Links

Alois P. Heinz, Rows n = 0..140, flattened

D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Formula

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

Example

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, ...

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 *)

Crossrefs

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

Sequence in context: A323873 A365582 A367559 * A179438 A211970 A089688

Adjacent sequences: A099017 A099018 A099019 * A099021 A099022 A099023

Keyword

nonn,tabl

Author

Ralf Stephan, Sep 23 2004

Status

approved