A360560
Triangle read by rows. T(n, k) = (1/2) * C(n, k) * C(3*n - 1, n) for n > 0 and T(0, 0) = 1.
Original entry on oeis.org
1, 1, 1, 5, 10, 5, 28, 84, 84, 28, 165, 660, 990, 660, 165, 1001, 5005, 10010, 10010, 5005, 1001, 6188, 37128, 92820, 123760, 92820, 37128, 6188, 38760, 271320, 813960, 1356600, 1356600, 813960, 271320, 38760, 245157, 1961256, 6864396, 13728792, 17160990, 13728792, 6864396, 1961256, 245157
Offset: 0
Triangle begins:
1;
1, 1;
5, 10, 5;
28, 84, 84, 28;
165, 660, 990, 660, 165;
1001, 5005, 10010, 10010, 5005, 1001;
-
T := (n, k) -> ifelse(n = 0, 1, binomial(n, k)*binomial(3*n - 1, n)/2):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
-
T(n,m):=1/2*binomial(n+1,m)*binomial(3*n+2,n+1);
A360282
Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.
Original entry on oeis.org
1, 0, 1, 0, 3, 2, 0, 10, 12, 3, 0, 35, 60, 30, 4, 0, 126, 280, 210, 60, 5, 0, 462, 1260, 1260, 560, 105, 6, 0, 1716, 5544, 6930, 4200, 1260, 168, 7, 0, 6435, 24024, 36036, 27720, 11550, 2520, 252, 8
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 3, 2;
[3] 0, 10, 12, 3;
[4] 0, 35, 60, 30, 4;
[5] 0, 126, 280, 210, 60, 5;
[6] 0, 462, 1260, 1260, 560, 105, 6;
[7] 0, 1716, 5544, 6930, 4200, 1260, 168, 7;
[8] 0, 6435, 24024, 36036, 27720, 11550, 2520, 252, 8;
[9] 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;
-
T := proc(n, k) if n = 0 then 1 else m := n - k + 1; (1/2) * binomial(2*m, m) * binomial(m + n - 1, k - 1) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..8);
# With Vladimir Kruchinin's g.f.:
gf := 1 + (y - x*y^2)/(2*sqrt((x*y - 1)^2 - 4*x)) ;
serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
seq(print(seq(coeff(poly(n), y, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 14 2023
Showing 1-2 of 2 results.
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