cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Mar 03 2009

Keywords

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

Examples

			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;
  ...

Links

Crossrefs

Cf. A007318, A022167, A157638, A157641.

Programs

  • 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
  • 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}]]
  • 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
  • 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

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

Extensions

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

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

Original entry on oeis.org

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

Views

Author

Roger L. Bagula, Mar 03 2009

Keywords

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

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
Showing 1-2 of 2 results.

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