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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).
0
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
0,2
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).
EXAMPLE
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;
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
PROG
(Maxima)
T(n, m):=if n<m then 0 else 1/(n+1)*binomial(n+1, m)*if evenp(n-m+1) then 4^(n)*binomial((3*n-m+1)/2-1, n) else binomial((3*n-m)/2, n)*binomial(3*n-m, (3*n-m)/2)/binomial(n-m, (n-m)/2);
CROSSREFS
Cf. A078531, A000984, A151403 (row sums).
Sequence in context: A366131 A307538 A316200 * A362745 A179789 A334396
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Feb 14 2023
STATUS
approved