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%I #11 May 10 2021 03:45:16
%S 1,1,1,1,1,1,1,1,1,1,1,5,5,5,1,1,5,25,25,5,1,1,9,45,225,45,9,1,1,25,
%T 225,1125,1125,225,25,1,1,29,725,6525,6525,6525,725,29,1,1,61,1769,
%U 44225,79605,79605,44225,1769,61,1,1,129,7869,228201,1141005,2053809,1141005,228201,7869,129,1
%N Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.
%C Start from the sequence 0, 1, 1, 1, 5, 5, 9, 25, 29, 61, 129, 177, 373, 693, 1081, 2185, 3853, ..., f(n) = f(n-2) + 4*f(n-3) and its partial products c(n) = 1, 1, 1, 1, 5, 25, 225, 5625, 163125, 9950625, ... . Then T(n,k) = round(c(n)/(c(k)*c(n-k))).
%H G. C. Greubel, <a href="/A172359/b172359.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-2, q) + q*f(n-3, q), f(0,q)=0, f(1,q) = f(2,q) = 1, and q = 4. - _G. C. Greubel_, May 09 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 1, 1, 1;
%e 1, 5, 5, 5, 1;
%e 1, 5, 25, 25, 5, 1;
%e 1, 9, 45, 225, 45, 9, 1;
%e 1, 25, 225, 1125, 1125, 225, 25, 1;
%e 1, 29, 725, 6525, 6525, 6525, 725, 29, 1;
%e 1, 61, 1769, 44225, 79605, 79605, 44225, 1769, 61, 1;
%e 1, 129, 7869, 228201, 1141005, 2053809, 1141005, 228201, 7869, 129, 1;
%t f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, 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, 4], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 09 2021 *)
%o (Sage)
%o @CachedFunction
%o def f(n,q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, 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,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 09 2021
%Y Cf. A172353 (q=1), A172358 (q=2), this sequence (q=4), A172360 (q=5).
%K nonn,tabl,less
%O 0,12
%A _Roger L. Bagula_, Feb 01 2010
%E Definition corrected to give integral terms by _G. C. Greubel_, May 09 2021
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