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Triangle T(n, k) = binomial(n+k, 2*k)*k!, read by rows.
2

%I #10 Jun 05 2021 06:36:16

%S 1,1,1,1,3,2,1,6,10,6,1,10,30,42,24,1,15,70,168,216,120,1,21,140,504,

%T 1080,1320,720,1,28,252,1260,3960,7920,9360,5040,1,36,420,2772,11880,

%U 34320,65520,75600,40320,1,45,660,5544,30888,120120,327600,604800,685440,362880

%N Triangle T(n, k) = binomial(n+k, 2*k)*k!, read by rows.

%H Harvey P. Dale, <a href="/A156367/b156367.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1 -x -x*y/(1 -x -x*y/(1 -x -2*x*y/(1 -x -2*x*y/(1 -x -3*x*y/(1 -x -3*x*y/(1 - ... (continued fraction).

%F T(n, k) = binomial(n+k, 2*k)*k!

%F T(n, k) = A155856(n, n-k).

%F Sum_{k=0..n} T(n, k) = A155857(n).

%F sum_{k=0..floor(n/2)} T(n, k) = A084261(n).

%e Triangle begins

%e 1;

%e 1, 1;

%e 1, 3, 2;

%e 1, 6, 10, 6;

%e 1, 10, 30, 42, 24;

%e 1, 15, 70, 168, 216, 120;

%e 1, 21, 140, 504, 1080, 1320, 720;

%e 1, 28, 252, 1260, 3960, 7920, 9360, 5040;

%e 1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320;

%t Flatten[Table[Binomial[n+k,2k]k!,{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jun 17 2015 *)

%o (Sage) flatten([[factorial(k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 05 2021

%Y Cf. A084261 (diagonal sums), A155856 (row reversal), A155857 (row sums)

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Feb 08 2009