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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.

%I A337994 #14 Nov 01 2020 10:42:05
%S A337994 1,0,2,0,3,15,0,6,30,84,0,14,70,196,420,0,36,180,504,1080,1980,0,99,
%T A337994 495,1386,2970,5445,9009,0,286,1430,4004,8580,15730,26026,40040,0,858,
%U A337994 4290,12012,25740,47190,78078,120120,175032
%N 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.
%C A337994 T(n, k) is divisible by A099398(n) for all 0 <= k <= n.
%F A337994 Let t(n) denote the triangular numbers and C(n) the Catalan numbers.
%F A337994 T(n, k) = k*(2*n - 1)*(t(2*k + 1)/t(n + 1))*C(n - 1) for n, k > 0.
%F A337994 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)).
%e A337994 Triangle starts:
%e A337994 [0] 1
%e A337994 [1] 0, 2
%e A337994 [2] 0, 3,    15
%e A337994 [3] 0, 6,    30,    84
%e A337994 [4] 0, 14,   70,    196,   420
%e A337994 [5] 0, 36,   180,   504,   1080,  1980
%e A337994 [6] 0, 99,   495,   1386,  2970,  5445,   9009
%e A337994 [7] 0, 286,  1430,  4004,  8580,  15730,  26026,  40040
%e A337994 [8] 0, 858,  4290,  12012, 25740, 47190,  78078,  120120, 175032
%e A337994 [9] 0, 2652, 13260, 37128, 79560, 145860, 241332, 371280, 541008, 755820
%p A337994 T := proc(n, k) if n = 0 then 1 else
%p A337994 (k*(2*k+2)*(2*k+1)*(2*n-1)*binomial(2*(n-1), n-1))/(n*(n+1)*(n+2)) fi end:
%p A337994 # Recursive:
%p A337994 CatalanNumber := n -> binomial(2*n, n)/(n+1):
%p A337994 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:
%p A337994 seq(seq(T(n, k), k=0..n), n=0..9);
%t A337994 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
%Y A337994 Cf. A119578 (row sums), (-1)^n*A005430 (alternating row sums), A002740 (main diagonal), A007054 (col 1), A099398 (universal divisor), A000108 (Catalan).
%K A337994 nonn,tabl
%O A337994 0,3
%A A337994 _Peter Luschny_, Nov 01 2020