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%I #9 Sep 08 2022 08:45:41
%S 1,1,1,1,15,1,1,155,155,1,1,1395,14415,1395,1,1,11811,1098423,1098423,
%T 11811,1,1,97155,76499847,688498623,76499847,97155,1,1,788035,
%U 5104102695,388932625359,388932625359,5104102695,788035,1,1,6347715
%N General q-Narayana triangle sequence: T(n, k) = Product_{j=0..2} ( q_binomial(n+j, j+k, 2)/q_binomial(n+j-k, j, 2) ).
%C The row sums are: {1, 2, 17, 312, 17207, 2220470, 841692629, 788075032180, 2188496700502675, 15489902235905315506, 328274267992545230058705, ...}.
%H G. C. Greubel, <a href="/A156939/b156939.txt">Rows n = 0..50 of triangle, flattened</a>
%F T(n, k) = Product_{j=0..2} ( q_binomial(n+j, j+k, 2)/q_binomial(n+j-k, j, 2) ). - _G. C. Greubel_, May 22 2019
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 15, 1;
%e 1, 155, 155, 1;
%e 1, 1395, 14415, 1395, 1;
%e 1, 11811, 1098423, 1098423, 11811, 1;
%e 1, 97155, 76499847, 688498623, 76499847, 97155, 1;
%t (* First Program *)
%t t[n_, m_]:= If[m==0, n!, Product[Sum[(m+1)^i, {i, 0, k-1}], {k, 1, n}]];
%t b[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])];
%t c[n_, l_, m_]:= Product[b[n+k, l+k, 1]/b[n-l+k, k, 1], {k, 0, m}];
%t Table[c[n, k, 2], {n,0,10}, {k,0,n}]//Flatten
%t (* Second Program *)
%t T[n_, k_]:= Product[QBinomial[n+j,j+k,2]/QBinomial[n+j-k,j,2], {j,0,2}];
%t Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 22 2019 *)
%o (PARI)
%o b(n,k,q) = prod(j=1, k, (1-q^(n-j+1))/(1-q^j));
%o T(n, k) = prod(j=0, 2, b(n+j, j+k, 2)/b(n-k+j,j,2));
%o for(n=0, 12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 22 2019
%o (Magma)
%o B:= func< n,k,q | (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >;
%o T:= func< n,k | k eq 0 select 1 else B(n,k,2)*(&*[B(n+j, j+k, 2)/B(n-k+j,j,2): j in [1..2]]) >;
%o [[T(n,k) : k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 22 2019
%o (Sage)
%o def T(n, k): return product((q_binomial(n+j, j+k, 2)/q_binomial(n+j-k, j, 2)) for j in (0..2))
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 22 2019
%Y Cf. A001263, A156916, A156917, this sequence.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 18 2009
%E Edited by _G. C. Greubel_, May 22 2019
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