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.

A244038 a(n) = 4^n * binomial(3*n/2,n).

Original entry on oeis.org

1, 6, 48, 420, 3840, 36036, 344064, 3325608, 32440320, 318704100, 3148873728, 31256180280, 311452237824, 3113596420200, 31213674823680, 313672599360720, 3158823892156416, 31870058661517860, 322076161553203200, 3259691964853493400, 33034843349204336640, 335189468043077792760
Offset: 0

Views

Author

N. J. A. Sloane, Jun 28 2014

Keywords

Links

Crossrefs

Cf. A045741, A244039.
Cf. A160906, A383326.
Cf. A000302, A098430, A386897.

Programs

  • Magma
    [Round(4^n*Gamma(3*n/2+1)/(Gamma(n+1)*Gamma(n/2+1))): n in [0..40]]; // G. C. Greubel, Aug 06 2018
  • Maple
    f1:=n->4^n*binomial(3*n/2,n); [seq(f1(n),n=0..40)];
  • Mathematica
    Table[4^n Binomial[3 n/2, n], {n, 0, 40}] (* Vincenzo Librandi, Jun 29 2014 *)
  • PARI
    {a(n) = if( n<1, n==0, 3^n * polcoeff( serreverse( x / (x+1) / 2 * sqrt((x+3) / (x+1) / 3 + x * O(x^n))), n))}; /* Michael Somos, Jan 27 2018 */

Formula

a(n) = A045741(n+1) + A244039(n) [Gessel].
a(n) = [x^n] 1/sqrt(1 - 4*x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017
G.f. A(x) satisfies: A(x)^3 * (1 - 108*x^2) = 3*A(x) - 2. - Michael Somos, Jan 27 2018
a(n) = [x^n] ( (1 + 4*x)^(3/2) )^n. It follows that the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022
G.f.: 2/(1-2*sin(arcsin(216*x^2-1)/3)). - Vladimir Kruchinin, Oct 06 2022
G.f.: ((3^(5/6)*i + 3^(1/3))*(-18*i*z + sqrt(-324*z^2 + 3))^(1/3) - (3^(5/6)*i - 3^(1/3))*(18*i*z + sqrt(-324*z^2 + 3))^(1/3))/(2*sqrt(-324*z^2 + 3)), where i = sqrt(-1) is the imaginary unit. - Karol A. Penson, Oct 24 2024
From Seiichi Manyama, Aug 07 2025: (Start)
a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x) * (1-2*x))^(n+1).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k,n-k). (End)

A386895 a(n) = Sum_{k=0..n} binomial(5*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 8, 94, 1220, 16590, 231808, 3297154, 47490696, 690461070, 10111370720, 148929775544, 2203898519732, 32741261744802, 488010179737920, 7294326822378060, 109294796958693520, 1641111255497600910, 24688289062391137056, 372020649062760239080, 5614219481885985162960
Offset: 0

Views

Author

Seiichi Manyama, Aug 07 2025

Keywords

Crossrefs

Cf. A386812, A386896, A386897, A386898.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(5*n+1, k)*binomial(2*n-k, n-k));

Formula

a(n) = [x^n] (1+x)^(5*n+1)/(1-x)^(n+1).
a(n) = [x^n] 1/((1-x)^(3*n+1) * (1-2*x)^(n+1)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n+1,k) * binomial(4*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(4*n-k,n-k).
a(n) = binomial(2*n, n)*hypergeom([-1-5*n, -n], [-2*n], -1). - Stefano Spezia, Aug 07 2025
D-finite with recurrence +135*n*(n-1)*(3*n-1)*(3*n-2)*a(n) +3*(n-1)*(104049*n^3 -434754*n^2 +745789*n -439424)*a(n-1) +36*(517211*n^4 -4353801*n^3 +13137926*n^2 -17477238*n +8846684)*a(n-2) +16*(-11442763*n^4 +46270475*n^3 +85309279*n^2 -584322689*n +652846590)*a(n-3) -4585920*(5*n-16) *(5*n-14) *(5*n-18)*(5*n-17)*a(n-4)=0. - R. J. Mathar, Aug 21 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

Cf. A386812, A386895, A386897, A386898.

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

Cf. A386812, A386895, A386896, A386897.
Cf. A178792, A383326, A383716.

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
Showing 1-4 of 4 results.

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

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