cp's OEIS Frontend

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).

%I A360636 #25 Feb 15 2023 22:42:47
%S A360636 1,2,2,10,16,6,64,140,96,20,462,1280,1260,512,70,3584,12012,15360,
%T A360636 9240,2560,252,29172,114688,180180,143360,60060,12288,924,245760,
%U A360636 1108536,2064384,2042040,1146880,360360,57344,3432
%N 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).
%F A360636 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).
%e A360636 Triangle T(n, m) begins:
%e A360636 [0]     1;
%e A360636 [1]     2,      2;
%e A360636 [2]    10,     16,      6;
%e A360636 [3]    64,    140,     96,     20;
%e A360636 [4]   462,   1280,   1260,    512,    70;
%e A360636 [5]  3584,  12012,  15360,   9240,  2560,   252;
%e A360636 [6] 29172, 114688, 180180, 143360, 60060, 12288, 924;
%p A360636 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;
%p A360636 # Alternative:
%p A360636 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
%o A360636 (Maxima)
%o A360636 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);
%Y A360636 Cf. A078531, A000984, A151403 (row sums).
%K A360636 nonn,tabl
%O A360636 0,2
%A A360636 _Vladimir Kruchinin_, Feb 14 2023