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.

A108478 Diagonal sums of number triangle A108477.

Original entry on oeis.org

1, 1, 2, 14, 43, 127, 468, 1596, 5253, 17917, 60918, 205194, 694287, 2351611, 7951336, 26894840, 91004105, 307854073, 1041410602, 3523170438, 11918842803, 40320750711, 136404504124, 461454010164, 1561085306061, 5281113937653
Offset: 0

Views

Author

Paul Barry, Jun 04 2005

Keywords

Programs

  • Maple
    A108478:=n->add(add(binomial(2*(n-2*k),j)*binomial(2*k,j)*2^j, j=0..n-k), k=0..floor(n/2)): seq(A108478(n), n=0..30); # Wesley Ivan Hurt, Sep 26 2014
  • Mathematica
    Table[Sum[Sum[Binomial[2 (n - 2 k), j]*Binomial[2 k, j]*2^j, {j, 0, n - k}], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Wesley Ivan Hurt, Sep 26 2014 *)
  • PARI
    a(n) = sum(k=0, n\2, sum(j=0, n-k, binomial(2*(n-2*k), j)*binomial(2*k, j)*2^j)); \\ Michel Marcus, Sep 26 2014

Formula

a(n) = sum_{k=0..floor(n/2)} ( sum_{j=0..n-k} C(2(n-2k), j)*C(2k, j)*2^j ).
Empirical g.f.: -(3*x^3+x^2+x-1) / ((x^3-3*x^2-x-1)*(x^3+x^2+3*x-1)). - Colin Barker, Sep 26 2014

A108475 Expansion of (1-3*x) / (1-5*x-5*x^2+x^3).

Original entry on oeis.org

1, 2, 15, 84, 493, 2870, 16731, 97512, 568345, 3312554, 19306983, 112529340, 655869061, 3822685022, 22280241075, 129858761424, 756872327473, 4411375203410, 25711378892991, 149856898154532, 873430010034205, 5090723162050694, 29670908962269963
Offset: 0

Views

Author

Paul Barry, Jun 04 2005

Keywords

Comments

Row sums of A108477. In general, Sum_{k=0..n} Sum_{j=0..n} binomial(2*(n-k), j)*binomial(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3).

Links

Programs

  • Mathematica
    CoefficientList[Series[(1-3x)/(1-5x-5x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{5,5,-1},{1,2,15},30] (* Harvey P. Dale, Dec 30 2019 *)
  • PARI
    Vec((1-3*x)/((1+x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 04 2016

Formula

G.f.: (1-3*x)/((1+x)*(1-6*x+x^2)).
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(2*(n-k), j)*binomial(2*k, j)*2^j.
Conjecture: a(n) = A000129(n+1)*A001333(n). - R. J. Mathar, Jul 08 2009
a(n) + a(n+1) = A001541(n+1). - R. J. Mathar, Jul 13 2009
a(n) = (4*(-1)^n - (3-2*sqrt(2))^n*(-2+sqrt(2)) + (2+sqrt(2))*(3+2*sqrt(2))^n)/8. - Colin Barker, Nov 04 2016
a(n) = (-1)^n * Re(sqrt(1+i) * cos((n + 1/2) * arccos(i)) * sin(n * arccos(i)) + 1), where i = sqrt(-1). - Daniel Suteu, Jun 23 2018
Showing 1-2 of 2 results.

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