A157641 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 4.

1, 1, 1, 1, 10, 1, 1, 63, 63, 1, 1, 340, 2142, 340, 1, 1, 1705, 57970, 57970, 1705, 1, 1, 8190, 1396395, 7536100, 1396395, 8190, 1, 1, 38227, 31307913, 847301455, 847301455, 31307913, 38227, 1, 1, 174760, 668055052, 86847156760, 435512947870

Offset: 0

Author

Roger L. Bagula, Mar 03 2009

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 12, 128, 2824, 119352, 10345272, 1757295192, 610543721016, 418465696229912, 584788183756728952,...}.

Other triangles in the family (see name) include: q = 2 (see A157638), q = 3 (see A157640), and q = 4 (this triangle). - Werner Schulte, Nov 16 2018

Examples

			Triangle begins:
  1;
  1, 1;
  1, 10, 1;
  1, 63, 63, 1;
  1, 340, 2142, 340, 1;
  1, 1705, 57970, 57970, 1705, 1;
  1, 8190, 1396395, 7536100, 1396395, 8190, 1;
  1, 38227, 31307913, 847301455, 847301455, 31307913, 38227, 1;
  ...
		

Links

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

Crossrefs

Cf. A007318, A022168, A157638, A157640.

Programs

Magma

q:=4; [[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

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, 3], {k, 0, n}], {n, 0, 10}]]

PARI

T(n, k) = {binomial(n, k)*polcoef(x^k/prod(j=0, k, 1-4^j*x+x*O(x^n)), n)} \\ Andrew Howroyd, Nov 19 2018

PARI

my(q=4); 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

Sage

[[ binomial(n,k)*gaussian_binomial(n,k).subs(q=4) for k in range(n+1)] for n in range(10)] # G. C. Greubel, Nov 17 2018

Formula

T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*4^i.

T(n,k) = binomial(n,k) * A022168(n,k) for 0 <= k <= n. - Werner Schulte, Nov 16 2018

Extensions

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