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) ).

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

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 A340430

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