A156917 General q-Narayana triangle sequence: T(n, k) = Product_{j=0..2} ( q_binomial(n+j, j+k, 3)/q_binomial(n+j-k, j, 3) ).

1, 1, 1, 1, 40, 1, 1, 1210, 1210, 1, 1, 33880, 1024870, 33880, 1, 1, 925771, 784128037, 784128037, 925771, 1, 1, 25095280, 580812061522, 16262737722616, 580812061522, 25095280, 1, 1, 678468820, 425659125229240, 325671796712891524

Offset

0,5

Comments

Row sums are: {1, 2, 42, 2422, 1092632, 1570107618, 17424412036222, 652194913033179170, 189060566695044668933610, ...}.

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, 3)/q_binomial(n+j-k, j, 3) ). - G. C. Greubel, May 22 2019

Example

Triangle begins as:
  1;
  1,        1;
  1,       40,            1;
  1,     1210,         1210,              1;
  1,    33880,      1024870,          33880,            1;
  1,   925771,    784128037,      784128037,       925771,        1;
  1, 25095280, 580812061522, 16262737722616, 580812061522, 25095280, 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, 2]/b[n-l+k, k, 2], {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, 3]/QBinomial[n+j-k, j, 3], {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, 3)/b(n-k+j, j, 3));
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, 3)*(&*[B(n+j, j+k, 3)/B(n-k+j, j, 3): 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, 3)/q_binomial(n+j-k, j, 3)) 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, this sequence, A156939.

Sequence in context: A013375 A013419 A013420 * A176644 A329337 A078084

Adjacent sequences: A156914 A156915 A156916 * A156918 A156919 A156920

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 18 2009

Extensions

Edited by G. C. Greubel, May 22 2019

Status

approved