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 1-5 of 5 results.

A337292 a(n) = 4*binomial(5*n,n)/(5*n-1).

Original entry on oeis.org

5, 20, 130, 1020, 8855, 81900, 791120, 7887660, 80560285, 838553320, 8863227100, 94871786100, 1026317094705, 11203116342560, 123243929011680, 1364973221797900, 15207477517956825, 170321264840835900, 1916512328325665070, 21655893753689280120
Offset: 1

Views

Author

Lucas A. Brown, Aug 21 2020

Keywords

Comments

a(n) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/4 are the path's endpoints.

Crossrefs

Cf. A002294, A118971, A337291.

Programs

  • Mathematica
    Array[4 Binomial[5 #, #]/(5 # - 1) &, 20] (* Michael De Vlieger, Aug 21 2020 *)
  • PARI
    a(n) = {4*binomial(5*n,n)/(5*n-1)} \\ Andrew Howroyd, Aug 21 2020

Formula

a(n) = 5*A118971(n-1).
G.f.: 5*x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
D-finite with recurrence 8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-6)*a(n-1)=0., a(0)=1. - R. J. Mathar, Jan 26 2025
G.f.: -4*4F3(-1/5,1/5,2/5,3/5; 1/4,1/2,3/4; 3125*x/256) . - R. J. Mathar, Aug 10 2025

A337350 a(n) is the number of lattice paths from (0,0) to (2n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (2k,2k).

Original entry on oeis.org

1, 6, 34, 300, 3146, 36244, 443156, 5646040, 74137050, 996217860, 13633173180, 189347631720, 2662142601924, 37815138677960, 541882155414376, 7823955368697776, 113712609033955834, 1662288563798703204, 24424940365489658540, 360537080085493670856
Offset: 0

Views

Author

Lucas A. Brown, Aug 24 2020

Keywords

Comments

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Links

Crossrefs

Cf. A000108, A001448, A337291, A337292, A337351, A337352.

Programs

  • PARI
    seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(4*k,2*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

Formula

G.f.: 2 - 1 / (Sum_{n>=0} binomial(4*n,2*n) * x^n).
a(n) = binomial(4*n,2*n) * (8*n+1) / (8*n^2 + 2*n - 1) for n >= 1. For proof, see the Quy Nhan link.
D-finite with recurrence n*(2*n+1)*(8*n-7)*a(n) -2*(4*n-5)*(4*n-3)*(8*n+1)*a(n-1)=0. - R. J. Mathar, Jan 26 2025
From Lucas A. Brown, Jul 13 2025: (Start)
G.f.: 2 - sqrt(2-32*x) / sqrt(1+sqrt(1-16*x)).
a(n) = A000108(2*n) + 4 * A000108(2*n-1). (End)

A337351 a(n) is the number of lattice paths from (0,0) to (3n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,2k).

Original entry on oeis.org

1, 10, 110, 1805, 34770, 731760, 16295600, 377438250, 8999246900, 219399101415, 5444124108810, 137040309706725, 3490834454580950, 89816746611096280, 2330761164942308080, 60932036847971297230, 1603218808449019802550, 42423276620326253035205
Offset: 0

Views

Author

Lucas A. Brown, Aug 24 2020

Keywords

Comments

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Links

Crossrefs

Cf. A337291, A337292, A337350, A337352.

Programs

  • PARI
    seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(5*k,2*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

Formula

G.f.: 2 - 1 / (Sum_{n>=0} binomial(5*n,2*n) * x^n).

A337352 a(n) is the number of lattice paths from (0,0) to (3n,3n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,3k).

Original entry on oeis.org

1, 20, 524, 19660, 854380, 40304080, 2004409236, 103440770760, 5486614131756, 297239307415792, 16376472734974384, 914734188877259884, 51680064605716043636, 2948046519564292501232, 169560941932509940657016, 9822377923336683964009296, 572554753384166308597716396
Offset: 0

Views

Author

Lucas A. Brown, Aug 24 2020

Keywords

Comments

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Links

Crossrefs

Cf. A337291, A337292, A337350, A337351.

Programs

  • PARI
    seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(6*k,3*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

Formula

G.f.: 2 - 1 / (Sum_{n>=0} binomial(6*n,3*n) * x^n).

A378503 Expansion of (Sum_{k>=0} binomial(4*k,k) * x^k)^3.

Original entry on oeis.org

1, 12, 132, 1396, 14436, 147120, 1483996, 14854968, 147821604, 1464031120, 14443875984, 142042418004, 1393053544508, 13630170286224, 133092301736232, 1297274743175856, 12624909478998948, 122692158505386960, 1190859983017752880, 11545524234978791952, 111820579340839270416
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2024

Keywords

Crossrefs

Cf. A005810, A078995, A337291, A378484.
Cf. A002457, A378483.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(4*k, k)*x^k)^3)

Formula

a(n) = Sum_{i+j+k=n, i,j,k >= 0} binomial(4*i,i) * binomial(4*j,j) * binomial(4*k,k).
G.f.: B(x)^3 where B(x) is the g.f. of A005810.
27*a(n) - 256*a(n-1) = 18*A005810(n) - A337291(n) for n > 0.
Showing 1-5 of 5 results.

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