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 1-6 of 6 results.

A365843 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^3 ).

Original entry on oeis.org

1, 6, 54, 578, 6810, 85278, 1113854, 15004746, 206955378, 2908113974, 41484917958, 599202514578, 8745727050762, 128790559374030, 1911191826600462, 28551332345784730, 429040549473424866, 6480799118506040934, 98349636147075506006, 1498732955394826784226
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Column k=3 of A378238.
Cf. A007297, A365842, A365844, A365845.
Cf. A032349, A365622, A365847.
Cf. A144097.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+k+2, k)*binomial(3*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(3*(n+1),n-k).
G.f.: B^3, where B is the g.f. of A144097.
a(n) ~ sqrt(8060 + 2651*sqrt(10)) * (223 + 70*sqrt(10))^n / (2 * sqrt(5*Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Nov 28 2024

A370102 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(5*n-k-1,n-k).

Original entry on oeis.org

1, 8, 128, 2312, 44032, 864008, 17282432, 350353928, 7172939776, 147972367880, 3070951360128, 64044689834760, 1341056098444288, 28176478479561992, 593725756425591680, 12542160174109922312, 265525958014053580800, 5632170795392966388744
Offset: 0

Views

Author

Seiichi Manyama, Feb 10 2024

Keywords

Crossrefs

Cf. A006318, A103885, A365847, A370098, A370099.
Cf. A001448, A370100, A370101.

Programs

  • Maple
    seq(simplify(binomial(5*n-1, n)*hypergeom([-n, -4*n], [1 - 5*n], -1)), n = 0..20); # Peter Bala, Jul 29 2024
  • PARI
    a(n) = sum(k=0, n, binomial(4*n, k)*binomial(5*n-k-1, n-k));

Formula

a(n) = [x^n] ( (1+x)^4/(1-x)^4 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ). See A365847.
From Peter Bala, Jul 20 2024: (Start)
a(n) = binomial(5*n-1, n)*hypergeom([-n, -4*n], [1 - 5*n], -1).
For n >=1, a(n) = (4/3) * [x^n] S(x)^(3*n) = (4/5) * [x^n] (1/S(-x))^(5*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the sequence of large Schröder numbers A006318.
n*(4*n - 3)*(2*n - 1)*(4*n - 1)*(85*n^4 - 510*n^3 + 1138*n^2 - 1119*n + 409)*a(n) = 2*(29665*n^8 - 237320*n^7 + 794282*n^6 - 1443212*n^5 + 1544750*n^4 - 987560*n^3 + 363568*n^2 - 69168*n + 5040)*a(n-1) + (n - 2)*(4*n - 7)*(2*n - 3)*(4*n - 5)*(85*n^4 - 170*n^3 + 118*n^2 - 33*n + 3)*a(n-2) with a(0) = 1 and a(1) = 8.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. (End)
a(n) ~ (349 + 85*sqrt(17))^n / (17^(1/4) * sqrt(Pi*n) * 2^(5*n - 1/2)). - Vaclav Kotesovec, Aug 08 2024
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] (1-x)^(n-1)/(1-2*x)^(4*n).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n,k) * binomial(n-1,n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+k-1,k) * binomial(n-1,n-k). (End)

A365846 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1+x)^4 ).

Original entry on oeis.org

1, 7, 73, 903, 12281, 177415, 2672377, 41506823, 659972089, 10689904647, 175765581817, 2925998735367, 49219210772473, 835307328307207, 14284937032826873, 245924997499453447, 4258621314671050745, 74128819286282600455, 1296324135131612708857
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A066357, A365754, A365847, A365848.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+k+2, k)*binomial(4*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(4*(n+1),n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^4 / (1-x)^3 )^(n+1). - Seiichi Manyama, Jul 31 2025

A365622 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^5 ).

Original entry on oeis.org

1, 10, 150, 2670, 52250, 1086002, 23533790, 525825830, 12026993010, 280220428890, 6627397194022, 158692955007390, 3839595257256330, 93725694152075010, 2305406918530451950, 57085385625207424342, 1421808255906105290210
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A032349, A365843, A365847.
Cf. A363006.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(5*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(5*(n+1),n-k).
G.f.: B^5, where B is the g.f. of A363006.

A365848 Expansion of (1/x) * Series_Reversion( x*(1-x)^5/(1+x)^4 ).

Original entry on oeis.org

1, 9, 122, 1965, 34814, 655290, 12861708, 260312853, 5393696150, 113847928558, 2439377254412, 52919446267698, 1160040801590332, 25655668799151700, 571760925292574640, 12827392114274902629, 289470689505615716070, 6566330844138035042982
Offset: 0

Views

Author

Seiichi Manyama, Sep 20 2023

Keywords

Crossrefs

Cf. A066357, A365754, A365846, A365847.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+k+4, k)*binomial(4*(n+1), n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(5*n+k+4,k) * binomial(4*(n+1),n-k).

A378840 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(4/3) * (1 + A(x)^(1/3)) )^3.

Original entry on oeis.org

1, 6, 66, 902, 13794, 225990, 3878946, 68854278, 1253647938, 23283474310, 439394508162, 8401507608966, 162413310158626, 3169029168475206, 62330703549363810, 1234503404283308038, 24599422679682518658, 492824963618477891334, 9920626149798702401730
Offset: 0

Views

Author

Seiichi Manyama, Dec 09 2024

Keywords

Crossrefs

Cf. A260332, A365847, A371676.
Cf. A006320, A033296, A365843.

Programs

  • PARI
    a(n, r=3, t=4, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

a(n) = 3 * Sum_{k=0..n} binomial(n,k) * binomial(4*n+k+3,n)/(4*n+k+3).
Conjecture: g.f.: B(x)^3, where B(x) is the g.f. of A260332.
Showing 1-6 of 6 results.

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