cp's OEIS Frontend

A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments.

%I A172368 #17 May 09 2021 09:52:19
%S A172368 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,1,1,5,15,15,15,5,1,1,7,35,
%T A172368 105,105,35,7,1,1,9,63,315,945,315,63,9,1,1,15,135,945,4725,4725,945,
%U A172368 135,15,1,1,25,375,3375,23625,39375,23625,3375,375,25,1
%N A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments.
%C A172368 Start from A052942 and its partial products c(n) = 1, 1, 1, 1, 1, 3, 15, 105, 945, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).
%H A172368 G. C. Greubel, <a href="/A172368/b172368.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A172368 T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = f(n-1, q) + q*f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 2. - _G. C. Greubel_, May 08 2021
%e A172368 Triangle begins as:
%e A172368   1;
%e A172368   1,  1;
%e A172368   1,  1,   1;
%e A172368   1,  1,   1,    1;
%e A172368   1,  1,   1,    1,     1;
%e A172368   1,  3,   3,    3,     3,     1;
%e A172368   1,  5,  15,   15,    15,     5,     1;
%e A172368   1,  7,  35,  105,   105,    35,     7,    1;
%e A172368   1,  9,  63,  315,   945,   315,    63,    9,   1;
%e A172368   1, 15, 135,  945,  4725,  4725,   945,  135,  15,  1;
%e A172368   1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1;
%t A172368 f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, f[n-1, q] + q*f[n-4, q]]];
%t A172368 c[n_, q_]:= Product[f[j, q], {j,n}];
%t A172368 T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
%t A172368 Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 08 2021 *)
%o A172368 (Sage)
%o A172368 @CachedFunction
%o A172368 def f(n,q): return 0 if (n==0) else 1 if (n<4) else f(n-1, q) + q*f(n-4, q)
%o A172368 def c(n,q): return product( f(j,q) for j in (1..n) )
%o A172368 def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
%o A172368 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 08 2021
%Y A172368 Cf. A052942, A172363 (q=1), this sequence (q=2), A172369 (q=3).
%K A172368 nonn,tabl,less
%O A172368 0,17
%A A172368 _Roger L. Bagula_, Feb 01 2010
%E A172368 Definition corrected to give integral terms, _G. C. Greubel_, May 08 2021