%I #14 Jun 28 2019 07:15:30

%S 1,4,2,16,16,5,64,96,60,14,256,512,480,224,42,1024,2560,3200,2240,840,

%T 132,4096,12288,19200,17920,10080,3168,429,16384,57344,107520,125440,

%U 94080,44352,12012,1430,65536,262144,573440,802816,752640,473088,192192,45760,4862

%N Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

%F T(n,0) = A000302(n).

%F T(n,n) = A000108(n+1).

%F T(n,1) = A002699(n) for n>=1.

%F T(n,n-1) = A128650(n+2) for n>=1.

%F T(2*n,n) = A254633(n).

%F T(n,k) = 4^(n-k)*C(n,k)*Catalan(k+1).

%F sum(k=0..n, T(n,k)) = A025230(n+2).

%e [ 1]

%e [ 4, 2]

%e [ 16, 16, 5]

%e [ 64, 96, 60, 14]

%e [ 256, 512, 480, 224, 42]

%e [1024, 2560, 3200, 2240, 840, 132]

%e [4096, 12288, 19200, 17920, 10080, 3168, 429]

%p h := n -> simplify(hypergeom([3/2, -n], [3], -x)):

%p seq(print(seq(4^n*coeff(h(n), x, k), k=0..n)), n=0..9);

%t T[n_, k_] := 4^(n-k) Binomial[n, k] CatalanNumber[k+1];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Jean-François Alcover_, Jun 28 2019 *)

%o (Sage)

%o A254632 = lambda n,k: (4)^(n-k)*binomial(n,k)*catalan_number(k+1)

%o for n in range(7): [A254632(n,k) for k in (0..n)]

%Y Cf. A108198 (Peter Bala), A000302, A000108, A025230, A002699, A128650, A254633.

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, Feb 03 2015