A156916 General q-Narayana triangle sequence: T(n, k) = Product_{j=0..1} (q-binomial(n+j, j+k, 2)/q-binomial(n-k+j, j, 2)).

1, 1, 1, 1, 7, 1, 1, 35, 35, 1, 1, 155, 775, 155, 1, 1, 651, 14415, 14415, 651, 1, 1, 2667, 248031, 1098423, 248031, 2667, 1, 1, 10795, 4112895, 76499847, 76499847, 4112895, 10795, 1, 1, 43435, 66982975, 5104102695, 21437231319, 5104102695, 66982975, 43435, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Formula

q=2; m=1; c(n,j,m) = Product_{k=0..m} (q-binomial(n + k, j + k, q)/q-binomial(n - j + k, k, q))

T(n,k) = Product_{i=1..k} (((2^(n+1-i)-1) / (2^i-1)) * ((2^(n+2-i)-1) / (2^(i+1)-1))) for 0 <= k <= n. - Werner Schulte, Nov 14 2018

T(n, k) = Product_{j=0..1} ( q_binomial(n+j, j+k, 2)/q_binomial(n+j-k, j, 2) ). - G. C. Greubel, May 22 2019

Example

Triangle begins:
  1;
  1,     1;
  1,     7,       1;
  1,    35,      35,        1;
  1,   155,     775,      155,        1;
  1,   651,   14415,    14415,      651,       1;
  1,  2667,  248031,  1098423,   248031,    2667,     1;
  1, 10795, 4112895, 76499847, 76499847, 4112895, 10795, 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, m]/b[n-l+k, k, m], {k, 0, m}];
Table[c[n, k, 1], {n, 0, 10}, {k, 0, n}]//Flatten
(* Second Program *)
m=1; q=2; Table[Product[QBinomial[n+k, k+j, q]/QBinomial[n+k-j, k, q], {k, 0, m}], {n, 0, 10}, {j, 0, n}]//Flatten (* G. C. Greubel, Nov 21 2018 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1( prod(i=1, k, ( (2^(n+1-i)-1)/(2^i-1) )*( (2^(n+2-i)-1)/(2^(i+1)-1)) ), ", "))) \\ G. C. Greubel, Nov 21 2018
(Magma) [[k le 0 select 1 else (&*[((2^(n+1-i)-1)/(2^i-1))*((2^(n+2-i) -1)/(2^(i+1)-1)): i in [1..k]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 21 2018
(Sage) [[prod(q_binomial(n+k, k+j, 2)/q_binomial(n+k-j, k, 2) for k in (0..1)) for j in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 21 2018

Crossrefs

Cf. A001263, this sequence, A156917, A156939.

Sequence in context: A154337 A033933 A108267 * A173584 A166973 A157156

Adjacent sequences: A156913 A156914 A156915 * A156917 A156918 A156919

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 18 2009

Extensions

Edited by G. C. Greubel, May 22 2019

Status

approved