A123382 Triangle T(n,k), 0 <= k <= n, defined by : T(n,k) = 0 if k < 0, T(0,k) = 0^k, (n+2)*(2*n-2*k+1)*T(n,k) = (2*n+1)*( 4*(2*n-2*k+1)*T(n-1,k-1) + (n+2*k+2)*T(n-1,k) ).

1, 1, 4, 1, 15, 20, 1, 35, 168, 112, 1, 66, 714, 1680, 672, 1, 110, 2178, 11352, 15840, 4224, 1, 169, 5434, 51051, 156156, 144144, 27456, 1, 245, 11830, 178035, 972400, 1953952, 1281280, 183040, 1, 340, 23324, 520676, 4516798, 16102944, 22870848, 11202048, 1244672, 1, 456, 42636, 1337220, 17073134

Offset: 0

Author

Philippe Deléham, Oct 13 2006

Keywords

• nonn
• tabl

Comments

G. Kreweras explains that since the rows of A140136 are symmetric, they can be considered as linear combinations of the odd-indexed rows of the Pascal triangle. For instance, (1,1) = 1*(1,1) and (1,7,7,1) = 1*(1,3,3,1) + 4*(0,1,1,0) and (1,20,75,75,10,1) = 1*(1,5,10,10,5,1) + 15*(0,1,3,3,1) + 20*(0,0,1,1,0,0). These coefficients (1; 1, 4; 1, 15, 20;) are the rows of this triangle. - Michel Marcus, Nov 17 2014

Examples

			Triangle begins:
0: 1;
1: 1, 4;
2: 1, 15, 20;
3: 1, 35, 168, 112;
4: 1, 66, 714, 1680, 672;
5: 1, 110, 2178, 11352, 15840, 4224;
6: 1, 169, 5434, 51051, 156156, 144144, 27456;
7: 1, 245, 11830, 178035, 972400, 1953952, 1281280, 183040;
8: 1, 340, 23324, 520676, 4516798, 16102944, 22870848, 11202048, 1244672;
.....
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965).
• Germain Kreweras, page 93 of "Sur une classe de problèmes de dénombrement...", containing the defining formula for this sequence.

Programs

Mathematica

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := T[n, k] = (2*n + 1)*(4*(2*n - 2*k + 1)*T[n - 1, k - 1] + (n + 2*k + 2)*T[n - 1, k])/((n + 2)*(2*n - 2*k + 1)); Table[If[k < 0, 0, T[n, k]], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 13 2017 *)

Sage

@CachedFunction
def T(n,k):
    if k < 0: return 0
    if n < 0: return 0
    if n == 0: return int( k==0 )
    if k == 0: return 1
    return ( (2*n+1)*( 4*(2*n-2*k+1)*T(n-1,k-1) + (n+2*k+2)*T(n-1,k) ) ) / ((n+2)*(2*n-2*k+1))
for n in [0..16]:
    print([T(n,k) for k in range(0,n+1)])
# Joerg Arndt, Nov 21 2014

Formula

T(n,n) = A003645(n).

Extensions

Corrected name, added more terms, Joerg Arndt, Nov 21 2014