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 1-3 of 3 results.

A134758 a(n) = A000984(n) + n.

Original entry on oeis.org

1, 3, 8, 23, 74, 257, 930, 3439, 12878, 48629, 184766, 705443, 2704168, 10400613, 40116614, 155117535, 601080406, 2333606237, 9075135318, 35345263819, 137846528840, 538257874461, 2104098963742, 8233430727623, 32247603683124, 126410606437777, 495918532948130
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Links

Crossrefs

Cf. A000984, A134759, A134760, A134761, A134762, A134763, A134770, A134771.

Programs

  • Magma
    [n+(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[Binomial[2n,n]+n,{n,0,40}] (* Harvey P. Dale, Dec 10 2011 *)
  • SageMath
    [n+binomial(2*n,n) for n in range(41)] # G. C. Greubel, May 28 2024

Formula

G.f.: ((1-x)^2 + x*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). - Harvey P. Dale, Dec 10 2011
From G. C. Greubel, May 28 2024: (Start)
E.g.f.: x*exp(x) + exp(2*x)*BesselI(0, 2*x).
a(n) = (2*(2*n-1)*a(n-1) - (3*n^2 - 6*n + 2))/n. (End)

Extensions

More terms from Harvey P. Dale, Dec 10 2011

A134759 a(n) = 2*A000984(n) - (n+1).

Original entry on oeis.org

1, 2, 9, 36, 135, 498, 1841, 6856, 25731, 97230, 369501, 1410852, 5408299, 20801186, 80233185, 310235024, 1202160763, 4667212422, 18150270581, 70690527580, 275693057619, 1076515748858, 4208197927417, 16466861455176, 64495207366175, 252821212875478
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Links

Crossrefs

Cf. A000108, A000984, A134758, A134760, A134761, A134762, A134763, A134770, A134771.

Programs

  • Magma
    [(n+1)*(2*Catalan(n)-1): n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[2 Binomial[2n,n]-n-1,{n,0,30}] (* Harvey P. Dale, Aug 07 2023 *)
  • SageMath
    [2*binomial(2*n,n) -(n+1) for n in range(41)] # G. C. Greubel, May 28 2024

Formula

From G. C. Greubel, May 28 2024: (Start)
a(n) = (n+1)*(2*A000108(n) - 1).
a(n) = (2*(2*n-1)*a(n-1) + 3*n*(n-1))/n.
G.f.: 2/sqrt(1-4*x) - 1/(1-x)^2.
E.g.f.: 2*exp(2*x)*BesselI(0, 2*x) - (1+x)*exp(x). (End)

Extensions

More terms from Harvey P. Dale, Aug 07 2023

A134762 a(n) = 3*A000984(n) - 2.

Original entry on oeis.org

1, 4, 16, 58, 208, 754, 2770, 10294, 38608, 145858, 554266, 2116294, 8112466, 31201798, 120349798, 465352558, 1803241168, 7000818658, 27225405898, 106035791398, 413539586458, 1614773623318, 6312296891158, 24700292182798, 96742811049298, 379231819313254
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Comments

Second inverse binomial transform of the sequence = A134763, (same as a(n) but with interpolated two's).

Links

Crossrefs

Cf. A000108, A000984, A134758, A134759, A134760, A134761, A134763, A134770, A134771.

Programs

  • Magma
    [3*(n+1)*Catalan(n)-2: n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[3*Binomial[2*n,n]-2, {n,0,40}] (* G. C. Greubel, May 28 2024 *)
  • PARI
    a(n) = 3*binomial(2*n, n) - 2; \\ Michel Marcus, Nov 22 2013
  • SageMath
    [3*binomial(2*n,n) -2 for n in range(41)] # G. C. Greubel, May 28 2024

Formula

G.f.: 3/sqrt(1-4*x) - 2/(1-x). - Sergei N. Gladkovskii, Nov 21 2013
From G. C. Greubel, May 28 2024: (Start)
a(n) = 3*(n+1)*A000108(n) - 2.
a(n) = (2*(2*n-1)*a(n-1) + 2*(3*n-2))/n.
E.g.f.: 3*exp(2*x)*BesselI(0, 2*x) - 2*exp(x). (End)

Extensions

More terms from Michel Marcus, Nov 22 2013
Showing 1-3 of 3 results.

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