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A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.

%I A254632 #14 Jun 28 2019 07:15:30
%S A254632 1,4,2,16,16,5,64,96,60,14,256,512,480,224,42,1024,2560,3200,2240,840,
%T A254632 132,4096,12288,19200,17920,10080,3168,429,16384,57344,107520,125440,
%U A254632 94080,44352,12012,1430,65536,262144,573440,802816,752640,473088,192192,45760,4862
%N A254632 Triangle read by rows, T(n, k) = 4^n*[x^k]hypergeometric([3/2, -n], [3], -x), n>=0, 0<=k<=n.
%F A254632 T(n,0) = A000302(n).
%F A254632 T(n,n) = A000108(n+1).
%F A254632 T(n,1) = A002699(n) for n>=1.
%F A254632 T(n,n-1) = A128650(n+2) for n>=1.
%F A254632 T(2*n,n) = A254633(n).
%F A254632 T(n,k) = 4^(n-k)*C(n,k)*Catalan(k+1).
%F A254632 sum(k=0..n, T(n,k)) = A025230(n+2).
%e A254632 [   1]
%e A254632 [   4,     2]
%e A254632 [  16,    16,     5]
%e A254632 [  64,    96,    60,    14]
%e A254632 [ 256,   512,   480,   224,    42]
%e A254632 [1024,  2560,  3200,  2240,   840,  132]
%e A254632 [4096, 12288, 19200, 17920, 10080, 3168, 429]
%p A254632 h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
%p A254632 seq(print(seq(4^n*coeff(h(n), x, k), k=0..n)), n=0..9);
%t A254632 T[n_, k_] := 4^(n-k) Binomial[n, k] CatalanNumber[k+1];
%t A254632 Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* _Jean-François Alcover_, Jun 28 2019 *)
%o A254632 (Sage)
%o A254632 A254632 = lambda n,k: (4)^(n-k)*binomial(n,k)*catalan_number(k+1)
%o A254632 for n in range(7): [A254632(n,k) for k in (0..n)]
%Y A254632 Cf. A108198 (Peter Bala), A000302, A000108, A025230, A002699, A128650, A254633.
%K A254632 nonn,tabl
%O A254632 0,2
%A A254632 _Peter Luschny_, Feb 03 2015