login
Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n,q) = Product_{j=1..n} (q^j -1)^(n-j) and q = 2, read by rows.
2

%I #6 Apr 26 2021 01:57:24

%S 1,1,1,1,1,1,1,3,3,1,1,21,63,21,1,1,315,6615,6615,315,1,1,9765,

%T 3075975,21531825,3075975,9765,1,1,615195,6007379175,630774813375,

%U 630774813375,6007379175,615195,1,1,78129765,48065040779175,156451707736214625,2346775616043219375,156451707736214625,48065040779175,78129765,1

%N Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n,q) = Product_{j=1..n} (q^j -1)^(n-j) and q = 2, read by rows.

%H G. C. Greubel, <a href="/A173503/b173503.txt">Rows n = 0..29 of the triangle, flattened</a>

%F T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n,q) = Product_{j=1..n} (q^j -1)^(n-j) and q = 2.

%e The triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 3, 3, 1;

%e 1, 21, 63, 21, 1;

%e 1, 315, 6615, 6615, 315, 1;

%e 1, 9765, 3075975, 21531825, 3075975, 9765, 1;

%e 1, 615195, 6007379175, 630774813375, 630774813375, 6007379175, 615195, 1;

%t c[n_, q_]:= Product[(q^m-1)^(n-m), {m,1,n}];

%t T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);

%t Table[T[n, k, 2], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 25 2021 *)

%o (Sage)

%o @CachedFunction

%o def c(n,q): return product( (q^j -1)^(n-j) for j in (1..n))

%o def T(n,k,q): return c(n,q)/(c(k,q)*c(n-k,q))

%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Apr 25 2021

%Y Cf. A173504, A173505.

%K nonn,tabl,less

%O 0,8

%A _Roger L. Bagula_, Feb 20 2010

%E Edited by _G. C. Greubel_, Apr 25 2021