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 A156939 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) ). 3
 1, 1, 1, 1, 15, 1, 1, 155, 155, 1, 1, 1395, 14415, 1395, 1, 1, 11811, 1098423, 1098423, 11811, 1, 1, 97155, 76499847, 688498623, 76499847, 97155, 1, 1, 788035, 5104102695, 388932625359, 388932625359, 5104102695, 788035, 1, 1, 6347715 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The row sums are: {1, 2, 17, 312, 17207, 2220470, 841692629, 788075032180, 2188496700502675, 15489902235905315506, 328274267992545230058705, ...}. LINKS G. C. Greubel, Rows n = 0..50 of triangle, flattened FORMULA 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 EXAMPLE Triangle begins as:   1;   1,     1;   1,    15,        1;   1,   155,      155,         1;   1,  1395,    14415,      1395,        1;   1, 11811,  1098423,   1098423,    11811,     1;   1, 97155, 76499847, 688498623, 76499847, 97155, 1; MATHEMATICA (* First Program *) t[n_, m_]:= If[m==0, n!, Product[Sum[(m+1)^i, {i, 0, k-1}], {k, 1, n}]]; b[n_, k_, m_]:= If[n==0, 1, t[n, m]/(t[k, m]*t[n-k, m])]; c[n_, l_, m_]:= Product[b[n+k, l+k, 1]/b[n-l+k, k, 1], {k, 0, m}]; Table[c[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten (* Second Program *) T[n_, k_]:= Product[QBinomial[n+j, j+k, 2]/QBinomial[n+j-k, j, 2], {j, 0, 2}]; Table[T[n, k], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, May 22 2019 *) PROG (PARI) b(n, k, q) = prod(j=1, k, (1-q^(n-j+1))/(1-q^j)); T(n, k) = prod(j=0, 2, b(n+j, j+k, 2)/b(n-k+j, j, 2)); for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, May 22 2019 (MAGMA) B:= func< n, k, q | (&*[(1-q^(n-j+1))/(1-q^j): j in [1..k]]) >; 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]]) >; [[T(n, k) : k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 22 2019 (Sage) 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)) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 22 2019 CROSSREFS Cf. A001263, A156916, A156917, this sequence. Sequence in context: A238754 A176226 A155493 * A174187 A174693 A022178 Adjacent sequences:  A156936 A156937 A156938 * A156940 A156941 A156942 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 18 2009 EXTENSIONS Edited by G. C. Greubel, May 22 2019 STATUS approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)