A360636 Triangle read by rows. T(n, m) = (1/(n + 1)) * C(n + 1, m) * 4^n * C((3*n - m + 1)/2 - 1, n) if n is odd, otherwise (1/(n + 1)) * C(n + 1, m) * C((3*n - m)/2, n) * C(3*n - m, (3*n - m)/2) / C(n - m, (n - m)/2).

1, 2, 2, 10, 16, 6, 64, 140, 96, 20, 462, 1280, 1260, 512, 70, 3584, 12012, 15360, 9240, 2560, 252, 29172, 114688, 180180, 143360, 60060, 12288, 924, 245760, 1108536, 2064384, 2042040, 1146880, 360360, 57344, 3432

Offset: 0

Author

Vladimir Kruchinin, Feb 14 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, m) begins:
[0]     1;
[1]     2,      2;
[2]    10,     16,      6;
[3]    64,    140,     96,     20;
[4]   462,   1280,   1260,    512,    70;
[5]  3584,  12012,  15360,   9240,  2560,   252;
[6] 29172, 114688, 180180, 143360, 60060, 12288, 924;
		

Crossrefs

Cf. A078531, A000984, A151403 (row sums).

Programs

Maple

T := (n, k) -> ifelse(n mod 2 = 1, 4^n*((3*n - k - 1)/2)! / (k!*(n + 1 - k)! * ((n - k - 1)/2)!), binomial(n + 1, k) * ((n - k)/2)! * (3*n - k)! / (((3*n - k)/2)! * (n + 1)! * (n - k)!)): for n from 0 to 6 do seq(simplify(T(n, k)), k=0..n) od;
# Alternative:
gf := ((1 - 4*x*y)*sin(arcsin((216*x^2) / (1 - 4*x*y)^3 - 1)/3))/(6*x) + (1 - 4*x*y) / (12*x): assume(x > 0); serx := series(gf, x, 9): poly := n -> simplify(coeff(serx, x, n)): seq(print(seq(coeff(poly(n), y, k), k = 0..n)), n = 0..6); # Peter Luschny, Feb 15 2023

Maxima

T(n,m):=if n
				

Formula

G.f.: ((1 - 4*x*y)*sin(arcsin((216*x^2) / (1 - 4*x*y)^3 - 1)/3))/(6*x) + (1 - 4*x*y) / (12*x).