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.

A386897 a(n) = 4^n * binomial(5*n/2,n).

Original entry on oeis.org

1, 10, 160, 2860, 53760, 1040060, 20500480, 409404600, 8255569920, 167718033340, 3427543285760, 70384350760360, 1451115518361600, 30018413447053080, 622759359440486400, 12951795276279787760, 269947721071617638400, 5637113741080428839100
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Sum[2^k *(-1)^(n-k)*Binomial[5*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
    A386897[n_] := 4^n*Binomial[5*n/2, n]; Array[A386897, 20, 0] (* Paolo Xausa, Aug 26 2025 *)
  • PARI
    a(n) = 4^n*binomial(5*n/2, n);

Formula

a(n) == 0 (mod 10) for n > 0.
a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(3*n+1).
a(n) = [x^n] 1/((1-x)^(n+1) * (1-2*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n+k,k) * binomial(2*n-k,n-k).
a(n) = [x^n] 1/(1-4*x)^(3*n/2+1).
a(n) = [x^n] (1+4*x)^(5*n/2).
a(n) ~ 2^(n - 1/2) * 5^((5*n+1)/2) / (sqrt(Pi*n) * 3^((3*n+1)/2)). - Vaclav Kotesovec, Aug 07 2025
D-finite with recurrence 3*n*(n-1)*(3*n-4) *(3*n-2)*a(n) -20*(5*n-4) *(5*n-8)*(5*n-2) *(5*n-6)*a(n-2)=0. - R. J. Mathar, Aug 21 2025
O.g.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (12500*x^2)/27) + 10*x*hypergeom([7/10, 9/10, 11/10, 13/10], [5/6, 7/6, 3/2], (12500*x^2)/27). - Karol A. Penson, Aug 26 2025

A386896 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 9, 125, 1932, 31365, 523809, 8910356, 153544680, 2671398309, 46822319115, 825501663525, 14623742203200, 260088366645900, 4641248247561324, 83059406374007720, 1490097583932329232, 26790218420643034533, 482571492068274975135, 8707190579448431827991
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(3*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(2*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(2*n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(3*n, n)*hypergeom([-1-5*n, -n], [-3*n], -1). - Stefano Spezia, Aug 07 2025
D-finite with recurrence 202*n*(n-1)*(2*n-1)*(2*n-3)*a(n) -3*(n-1)*(2*n-3) *(14093*n^2-15245*n+5226)*a(n-1) +4*(355081*n^4 -1597876*n^3 +2789549*n^2 -2405270*n+926160)*a(n-2) -3840*(5*n-11)*(5*n-9) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Aug 21 2025

A386898 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(5*n-k,n-k).

Original entry on oeis.org

1, 11, 199, 4031, 85919, 1885311, 42154111, 955020287, 21847988735, 503573013503, 11675986431999, 272033089535999, 6363380561141759, 149354395882487807, 3515589114309115903, 82957940541503045631, 1961823306198598418431, 46482660516543479939071
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(4*n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x)^(4*n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(4*n+k,k).
a(n) = binomial(5*n, n)*hypergeom([-1-5*n, -n], [-5*n], -1). - Stefano Spezia, Aug 07 2025

A385639 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 7, 69, 748, 8485, 98847, 1171884, 14066808, 170421669, 2079531685, 25520363869, 314653207128, 3894577133356, 48362609654548, 602248101550920, 7517853111444528, 94044248726758821, 1178641094940246897, 14796230460187072719, 186022053254555479500, 2341837809478393341885
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - Vaclav Kotesovec, Aug 07 2025
Showing 1-4 of 4 results.