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 1-4 of 4 results.

A162481 Expansion of (1/(1-x)^3)*c(x/(1-x)^3), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 4, 14, 54, 235, 1119, 5658, 29800, 161621, 896198, 5056824, 28938519, 167548937, 979653821, 5776252440, 34305807512, 205039491091, 1232333298174, 7443336041318, 45157243590384, 275051410542141, 1681362181696823, 10311616254855422, 63428758470722109
Offset: 0

Views

Author

Paul Barry, Jul 04 2009

Keywords

Links

Crossrefs

Cf. A000108, A086616, A162482, A360045.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n + 2*k + 2, n - k] * CatalanNumber[k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jun 30 2020 *)

Formula

G.f.: 1/((1-x)^3-x-x^2/((1-x)^3-2*x-x^2/((1-x)^3-2*x-x^2/((1-x)^3-2*x-x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..n} C(n+2k+2,n-k)*A000108(k).
Conjecture: (n+1)*a(n) +2*(1-4*n)*a(n-1) +2*(5*n-3)*a(n-2) +4*(2-n)*a(n-3) +(n-3)*a(n-4) = 0. - R. J. Mathar, Dec 11 2011
G.f. A(x) satisfies: A(x) = 1/(1 - x)^3 + x * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020
a(n) = binomial(n+2,2) + Sum_{k=0..n-1} a(k) * a(n-k-1). - Seiichi Manyama, Jan 23 2023
G.f.: (1 - sqrt(1 - 4*x/(1-x)^3))/(2*x). - Vaclav Kotesovec, Jan 24 2023

A360046 a(n) = Sum_{k=0..floor(n/4)} binomial(n+3,4*k+3) * Catalan(k).

Original entry on oeis.org

1, 4, 10, 20, 36, 64, 120, 240, 497, 1036, 2158, 4524, 9625, 20816, 45560, 100368, 221915, 492380, 1097302, 2457228, 5526666, 12474000, 28233600, 64061920, 145704327, 332174532, 758977386, 1737703780, 3985847284, 9157908736, 21074460512, 48569746368, 112096071675
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Links

Crossrefs

Cf. A086615, A360045, A360047.
Cf. A000108.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n+3, 4*k+3)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2/((1-x)^2*((1-x)^2+sqrt((1-x)^4-4*x^4))))

Formula

a(n) = binomial(n+3,3) + Sum_{k=0..n-4} a(k) * a(n-k-4).
G.f. A(x) satisfies: A(x) = 1/(1-x)^4 + x^4 * A(x)^2.
G.f.: 2 / ( (1-x)^2 * ((1-x)^2 + sqrt((1-x)^4 - 4*x^4)) ).
D-finite with recurrence (n+4)*a(n) +5*(-n-3)*a(n-1) +10*(n+2)*a(n-2) +10*(-n-1)*a(n-3) +(n+8)*a(n-4) +3*(n-1)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

A360047 a(n) = Sum_{k=0..floor(n/5)} binomial(n+4,5*k+4) * Catalan(k).

Original entry on oeis.org

1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3005, 6400, 13500, 28050, 57800, 119515, 250425, 533525, 1151975, 2504700, 5453176, 11856275, 25748450, 55962300, 121981725, 266968938, 586630515, 1292992795, 2855288480, 6311930460, 13963767356, 30919563310
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Links

Crossrefs

Cf. A086615, A360045, A360046.
Cf. A000108.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n+4, 5*k+4)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(2/((1-x)^2*((1-x)^3+sqrt((1-x)^6-4*x^5*(1-x)))))

Formula

a(n) = binomial(n+4,4) + Sum_{k=0..n-5} a(k) * a(n-k-5).
G.f. A(x) satisfies: A(x) = 1/(1-x)^5 + x^5 * A(x)^2.
G.f.: 2 / ( (1-x)^2 * ((1-x)^3 + sqrt((1-x)^6 - 4*x^5*(1-x))) ).
D-finite with recurrence (n+5)*a(n) +6*(-n-4)*a(n-1) +15*(n+3)*a(n-2) +20*(-n-2)*a(n-3) +15*(n+1)*a(n-4) +10*(-n+1)*a(n-5) +5*(n-1)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

A364625 G.f. satisfies A(x) = 1/(1-x)^3 + x^2*A(x)^2.

Original entry on oeis.org

1, 3, 7, 16, 38, 95, 249, 678, 1901, 5451, 15906, 47066, 140868, 425657, 1296665, 3977684, 12276617, 38094013, 118768915, 371875752, 1168843808, 3686549845, 11664123048, 37011249678, 117750111763, 375529083267, 1200327617200, 3844662925222, 12338289374046
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Crossrefs

Cf. A000108, A086615, A162481, A360045.
Cf. A364626, A364627.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1-4*x^2/(1-x)^3))))

Formula

G.f.: A(x) = 2 / ( (1-x)^3 * (1 + sqrt( 1 - 4*x^2/(1-x)^3 )) ).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k+2,3*k+2) * binomial(2*k,k) / (k+1).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.