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A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments. 2

%I #17 May 09 2021 09:52:19

%S 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 105,105,35,7,1,1,9,63,315,945,315,63,9,1,1,15,135,945,4725,4725,945,

%U 135,15,1,1,25,375,3375,23625,39375,23625,3375,375,25,1

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

%C 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 G. C. Greubel, <a href="/A172368/b172368.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1, 1, 1;

%e 1, 1, 1, 1, 1;

%e 1, 3, 3, 3, 3, 1;

%e 1, 5, 15, 15, 15, 5, 1;

%e 1, 7, 35, 105, 105, 35, 7, 1;

%e 1, 9, 63, 315, 945, 315, 63, 9, 1;

%e 1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1;

%e 1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1;

%t f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, f[n-1, q] + q*f[n-4, q]]];

%t c[n_, q_]:= Product[f[j, q], {j,n}];

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

%t Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 08 2021 *)

%o (Sage)

%o @CachedFunction

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

%o def T(n,k,q): return round(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..12)]) # _G. C. Greubel_, May 08 2021

%Y Cf. A052942, A172363 (q=1), this sequence (q=2), A172369 (q=3).

%K nonn,tabl,less

%O 0,17

%A _Roger L. Bagula_, Feb 01 2010

%E Definition corrected to give integral terms, _G. C. Greubel_, May 08 2021

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)