A360667 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.

1, 1, 1, 2, 4, 2, 10, 30, 30, 10, 64, 256, 384, 256, 64, 462, 2310, 4620, 4620, 2310, 462, 3584, 21504, 53760, 71680, 53760, 21504, 3584, 29172, 204204, 612612, 1021020, 1021020, 612612, 204204, 29172, 245760, 1966080, 6881280, 13762560, 17203200, 13762560, 6881280, 1966080, 245760

Offset: 0

Author

Vladimir Kruchinin, Feb 16 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, m) starts:
[0] 1;
[1] 1,     1;
[2] 2,     4,      2;
[3] 10,    30,     30,       10;
[4] 64,    256,    384,      256,    64;
[5] 462,   2310,   4620,     4620,   2310,    462;
[6] 3584,  21504,  53760,    71680,  53760,   21504,    3584;
[7] 29172, 204204, 612612,   1021020,1021020, 612612,   204204,  29172;
		

Crossrefs

Cf. A078531.

Programs

Mathematica

T[0, 0] = 1;
T[n_, m_] := 4^(n-1)*Binomial[n, m]*Binomial[3n/2-2, n-1]/n;
Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 16 2023 *)

Maxima

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);

Formula

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

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