A236934 Triangle of Poupard numbers g_n(k) read by rows, n>=1, 1<=k<=2n-1.

1, 0, 2, 0, 0, 4, 8, 4, 0, 0, 32, 64, 80, 64, 32, 0, 0, 544, 1088, 1504, 1664, 1504, 1088, 544, 0, 0, 15872, 31744, 45440, 54784, 58112, 54784, 45440, 31744, 15872, 0, 0, 707584, 1415168, 2059264, 2576384, 2911744, 3027968, 2911744, 2576384, 2059264, 1415168, 707584, 0

Offset

1,3

Links

Peter Luschny, Row(n) for n = 1..25

Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.

Foata, Dominique; Han, Guo-Niu; Strehl, Volker The Entringer-Poupard matrix sequence. Linear Algebra Appl. 512, 71-96 (2017).

Christiane Poupard, Deux propriétés des arbres binaires ordonnés stricts, Europ. J. Combin., vol. 10, 1989, p. 369-374.

Formula

4^(-n)*sum(k=1..2*n+1, binomial(2*n,k-1)*T(n+1,k)) = A000364(n), n>=0. - Peter Luschny, May 11 2014

(-1)^n*sum(k=1..2*n+1, (-1)^(k-1)*binomial(2*n,k-1)*T(n+1,k)) = A000302(n), n>=0. - Peter Luschny, May 11 2014

Example

Triangle begins:
1,
0, 2, 0,
0, 4, 8, 4, 0,
0, 32, 64, 80, 64, 32, 0,
0, 544, 1088, 1504, 1664, 1504, 1088, 544, 0,
...

Maple

T := proc(n, k) option remember; local j;
  if n = 1 then 1
elif k = 1 then 0
elif k = 2 then 2*add(T(n-1, j), j=1..2*n-3)
elif k > n then T(n, 2*n-k)
else 2*T(n, k-1)-T(n, k-2)-4*T(n-1, k-2)
  fi end:
seq(print(seq(T(n, k), k=1..2*n-1)), n=1..6); # Peter Luschny, May 11 2014

Mathematica

T[n_, k_] /; 1 <= k <= 2n-1 := T[n, k] = Which[n == 1, 1, k == 1, 0, k == 2, 2 Sum[T[n-1, j], {j, 1, 2n-3}], k > n, T[n, 2n-k], True, 2 T[n, k-1] - T[n, k-2] - 4 T[n-1, k-2]]; T[_, _] = 0;
Table[T[n, k], {n, 1, 7}, {k, 1, 2n-1}] // Flatten (* Jean-François Alcover, Jul 08 2019, from Maple *)

Crossrefs

Cf. A000182 (row sums), A008282, A125053.

Sequence in context: A352450 A098699 A021837 * A155719 A153772 A197007

Adjacent sequences: A236931 A236932 A236933 * A236935 A236936 A236937

Keyword

nonn,tabf

Author

N. J. A. Sloane, Feb 17 2014

Extensions

More terms from Peter Luschny, May 11 2014

Status

approved