%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