A360667 - OEIS

%I #12 Feb 16 2023 11:59:04

%S 1,1,1,2,4,2,10,30,30,10,64,256,384,256,64,462,2310,4620,4620,2310,

%T 462,3584,21504,53760,71680,53760,21504,3584,29172,204204,612612,

%U 1021020,1021020,612612,204204,29172,245760,1966080,6881280,13762560,17203200,13762560,6881280,1966080,245760

%N Triangle read by rows: T(n,m)=4^(n-1)*C(n,m)*C(3*n/2-2,n-1)/n, for 0 <= m <= n, with T(0,0)=1.

%F G.f.: sin(arcsin(216*x^2*(y+1)^2-1)/3)/6+13/12.

%F G.f.: 1+x*(sqrt(3)/2)*(sech(arccosh(-sqrt(108)*x*(1+y))/3))*(1+y).

%e Triangle T(n, m) starts:

%e [0] 1;

%e [1] 1,     1;

%e [2] 2,     4,      2;

%e [3] 10,    30,     30,       10;

%e [4] 64,    256,    384,      256,    64;

%e [5] 462,   2310,   4620,     4620,   2310,    462;

%e [6] 3584,  21504,  53760,    71680,  53760,   21504,    3584;

%e [7] 29172, 204204, 612612,   1021020,1021020, 612612,   204204,  29172;

%t T[0, 0] = 1;

%t T[n_, m_] := 4^(n-1)*Binomial[n, m]*Binomial[3n/2-2, n-1]/n;

%t Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 16 2023 *)

%o (Maxima)

%o T(n,m):=if n=0 and m=0 then 1 else if n=0 then 0 else (4^(n-1)*binomial(n,m)*binomial((3*n)/2-2,n-1))/(n);

%Y Cf. A078531.

%K nonn,tabl

%O 0,4

%A _Vladimir Kruchinin_, Feb 16 2023