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A344565 Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).

%I A344565 #12 May 29 2021 05:55:45
%S A344565 3,15,15,45,90,45,105,315,315,105,210,840,1260,840,210,378,1890,3780,
%T A344565 3780,1890,378,630,3780,9450,12600,9450,3780,630,990,6930,20790,34650,
%U A344565 34650,20790,6930,990,1485,11880,41580,83160,103950,83160,41580,11880,1485
%N A344565 Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
%F A344565 T(n, k) = (n + 4)! / (8 * k! * (n - k)!).
%e A344565 Triangle begins:
%e A344565 [0]   3;
%e A344565 [1]  15,     15;
%e A344565 [2]  45,     90,    45;
%e A344565 [3]  105,   315,   315,    105;
%e A344565 [4]  210,   840,  1260,    840,    210;
%e A344565 [5]  378,  1890,  3780,   3780,   1890,    378;
%e A344565 [6]  630,  3780,  9450,  12600,   9450,   3780,    630;
%e A344565 [7]  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;
%e A344565 [8] 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;
%e A344565 [9] 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145.
%p A344565 T := (n, k) -> (n + 4)! / (8 * k! * (n - k)!):
%p A344565 for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%t A344565 T[n_, k_] := (n + 4)!/(8*k!*(n - k)!); Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 28 2021 *)
%Y A344565 Apparently a subtriangle of A344678. Row sums A344564.
%K A344565 nonn,tabl
%O A344565 0,1
%A A344565 _Peter Luschny_, May 28 2021