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.

A188482 Diagonal sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A188481).

Original entry on oeis.org

1, 4, 17, 71, 295, 1221, 5040, 20761, 85380, 350659, 1438568, 5896098, 24145941, 98812861, 404118745, 1651811920, 6748282361, 27556753703, 112482005583, 458958881572, 1872034052651, 7633342954234, 31116252892098, 126806214027741, 516633711969649
Offset: 0

Views

Author

Emanuele Munarini, Apr 01 2011

Keywords

Crossrefs

Cf. A141223, A188481.

Programs

  • Mathematica
    Table[Sum[Binomial[k,i]Binomial[2n+k-2i+2,n-k](-1)^i,{k,0,n},{i,0,k}],{n,0,12}]
  • Maxima
    makelist(sum(sum(binomial(k,i)*binomial(2*n+k-2*i+2,n-k)*(-1)^i,i,0,k),k,0,n),n,0,12);

Formula

a(n) = [x^n] 1/((1-x)^(n+1)*(1-2*x-x^2+x^3)).
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(k,i)*binomial(2*n+k-2*i+2, n-k)*(-1)^i.
G.f.: (2 - 7*x - 4*x^2 + x*sqrt(1-4*x))/(2 - 14*x + 16*x^2 + 30*x^3 + 8*x^4).
Conjecture: (-n+1)*a(n) + (7*n-9)*a(n-1) + 2*(-4*n+7)*a(n-2) + (-15*n+23)*a(n-3) + 2*(-2*n+3)*a(n-4) = 0. - R. J. Mathar, Jun 14 2016

A141223 Expansion of 1/(sqrt(1-4*x)*(1-3*x*c(x))), where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 5, 24, 113, 526, 2430, 11166, 51105, 233190, 1061510, 4822984, 21879786, 99135076, 448707992, 2029215114, 9170247393, 41416383366, 186957126702, 843575853984, 3804927658878, 17156636097156, 77339426905812, 348553445817084, 1570548863858778, 7075531788285276
Offset: 0

Views

Author

Paul Barry, Jun 14 2008

Keywords

Comments

Binomial transform of A126932. Hankel transform is (-1)^n.
Row sums of the Riordan matrix (1/(1-4*x),(1-sqrt(1-4*x))/(2*sqrt(1-4*x))) (A188481). - Emanuele Munarini, Apr 01 2001

Links

Crossrefs

Cf. A000108.
Cf. A000302, A000984, A026641.
Cf. A006256, A385605, A385632.

Programs

  • Mathematica
    CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2),{x,0,100}],x] (* Emanuele Munarini, Apr 01 2011 *)
  • Maxima
    makelist(sum(binomial(n+k,k)*3^(n-k),k,0,n),n,0,12); /* Emanuele Munarini, Apr 01 2011 */

Formula

a(n) = Sum_{k=0..n} C(2*n-k,n-k)*3^k.
From Emanuele Munarini, Apr 01 2011: (Start)
a(n) = [x^n] 1/((1-x)^(n+1) * (1-3*x)). [Corrected by Seiichi Manyama, Aug 03 2025]
a(n) = 3^(2*n+1)/2^(n+2) + (1/4)*Sum_{k=0..n} binomial(2*k,k)*(9/2)^(n-k).
D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.
G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)
G.f.: (1/(1-4*x)^(1/2)+3)/(4-18*x) = (2 + x/(Q(0)-2*x))/(2-9*x) where Q(k) = 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 18 2013
a(n) ~ 3^(2*n + 1) / 2^(n + 1). - Vaclav Kotesovec, Sep 15 2021
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+1,k). - Seiichi Manyama, Aug 03 2025
a(n) = 3^(2*n+1)*2^(-n-1) - binomial(2*n+1, n)*(hypergeom([1, -1-n], [1+n], -1/2) - 1). - Stefano Spezia, Aug 05 2025
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k). - Seiichi Manyama, Aug 07 2025
Showing 1-2 of 2 results.

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