A337994 T(n, k) = (k*(2*k+2)*(2*k+1)*(2*n-1)*binomial(2*(n-1),n-1))/(n*(n+1)*(n+2)) for n, k > 0 and T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

1, 0, 2, 0, 3, 15, 0, 6, 30, 84, 0, 14, 70, 196, 420, 0, 36, 180, 504, 1080, 1980, 0, 99, 495, 1386, 2970, 5445, 9009, 0, 286, 1430, 4004, 8580, 15730, 26026, 40040, 0, 858, 4290, 12012, 25740, 47190, 78078, 120120, 175032

Offset: 0

Author

Peter Luschny, Nov 01 2020

Keywords

• nonn
• tabl

Comments

T(n, k) is divisible by A099398(n) for all 0 <= k <= n.

Examples

			Triangle starts:
[0] 1
[1] 0, 2
[2] 0, 3,    15
[3] 0, 6,    30,    84
[4] 0, 14,   70,    196,   420
[5] 0, 36,   180,   504,   1080,  1980
[6] 0, 99,   495,   1386,  2970,  5445,   9009
[7] 0, 286,  1430,  4004,  8580,  15730,  26026,  40040
[8] 0, 858,  4290,  12012, 25740, 47190,  78078,  120120, 175032
[9] 0, 2652, 13260, 37128, 79560, 145860, 241332, 371280, 541008, 755820
		

Crossrefs

Cf. A119578 (row sums), (-1)^n*A005430 (alternating row sums), A002740 (main diagonal), A007054 (col 1), A099398 (universal divisor), A000108 (Catalan).

Programs

Maple

T := proc(n, k) if n = 0 then 1 else
(k*(2*k+2)*(2*k+1)*(2*n-1)*binomial(2*(n-1), n-1))/(n*(n+1)*(n+2)) fi end:
# Recursive:
CatalanNumber := n -> binomial(2*n, n)/(n+1):
T := proc(n, k) option remember; if k=0 then k^n elif k=n then CatalanNumber(n+1)* binomial(n+1, 2) else (4 - 10/(n + 2))*T(n-1, k) fi end:
seq(seq(T(n, k), k=0..n), n=0..9);

Mathematica

T[n_, k_] := If[n == 0, 1, (k (2k + 2)(2k + 1)(2n - 1) CatalanNumber[n-1])/((n + 1) (n + 2))]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Formula

Let t(n) denote the triangular numbers and C(n) the Catalan numbers.

T(n, k) = k*(2*n - 1)*(t(2*k + 1)/t(n + 1))*C(n - 1) for n, k > 0.

T(n, k) = k^n if k = 0; if k = n then C(n+1)*t(n+1); else T(n-1, k)*(4-10/(n+2)).