A157640 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 3.

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 160, 780, 160, 1, 1, 605, 12100, 12100, 605, 1, 1, 2184, 165165, 677600, 165165, 2184, 1, 1, 7651, 2088723, 32401985, 32401985, 2088723, 7651, 1, 1, 26240, 25095280, 1405335680, 5313925540, 1405335680, 25095280

Offset

0,5

Comments

Row sums are: {1, 2, 10, 80, 1102, 25412, 1012300, 68996720, 8174839942, 1670428649564, 594362629986268,...}.

Other triangles in the family (see name) include: q = 2 (see A157638), q = 3 (this triangle), and q = 4 (see A157641). - Werner Schulte, Nov 16 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (rows n=0..49)

Formula

T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*3^i.

T(n,k) = binomial(n,k) * A022167(n,k) for 0 <= k <= n. - Werner Schulte, Nov 16 2018

Example

Triangle begins:
  1;
  1, 1;
  1, 8, 1;
  1, 39, 39, 1;
  1, 160, 780, 160, 1;
  1, 605, 12100, 12100, 605, 1;
  1, 2184, 165165, 677600, 165165, 2184, 1;
  1, 7651, 2088723, 32401985, 32401985, 2088723, 7651, 1;
  1, 26240, 25095280, 1405335680, 5313925540, 1405335680, 25095280, 26240, 1;
  ...

Mathematica

t[n_, m_] = Product[Sum[k*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}];
b[n_, k_, m_] = t[n, m]/(t[k, m]*t[n - k, m]);
Flatten[Table[Table[b[n, k, 2], {k, 0, n}], {n, 0, 10}]]

Prog

(PARI) T(n, k) = {binomial(n, k)*polcoef(x^k/prod(j=0, k, 1-3^j*x+x*O(x^n)), n)} \\ Andrew Howroyd, Nov 19 2018
(PARI) my(q=3); for(n=0, 10, for(k=0, n, print1(binomial(n, k)*prod(j=0, k-1, (1-q^(n-j))/(1-q^(j+1))), ", ")); print) \\ G. C. Greubel, Nov 17 2018
(Magma) q:=3; [[k le 0 select 1 else Binomial(n, k)*(&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 17 2018
(Sage) [[ binomial(n, k)*gaussian_binomial(n, k).subs(q=3) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 17 2018

Crossrefs

Cf. A007318, A022167, A157638, A157641.

Sequence in context: A156137 A152972 A166346 * A142458 A174528 A259465

Adjacent sequences: A157637 A157638 A157639 * A157641 A157642 A157643

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 03 2009

Extensions

Edited and simpler name by Werner Schulte and Andrew Howroyd, Nov 19 2018

Status

approved