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.

A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 6, 34, 188, 1026, 5556, 29940, 160824, 862018, 4613636, 24667644, 131795912, 703812916, 3757135752, 20051429544, 106992663408, 570827898306, 3045193326372, 16244056119084, 86646747723048, 462161936699196, 2465043081687192, 13147597801986264, 70123266087502608
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A293490, A383832, A386942.
Cf. A000302, A000984, A026641, A141223, A386957.

Programs

  • Magma
    [&+[3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[3^k * Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-4*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) ~ 2^(4*n+2) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

A383832 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n+2,n-k).

Original entry on oeis.org

1, 10, 78, 548, 3630, 23148, 143724, 874888, 5245038, 31065500, 182189348, 1059775608, 6122246572, 35160205752, 200902089240, 1142857957392, 6475994731758, 36569545322364, 205869970843764, 1155749458070040, 6472151016349284, 36161680227612456, 201628061114911848
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A293490, A377011, A386942.

Programs

  • Magma
    [&+[(k+1) * 3^k * Binomial(2*n+2,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[(k+1)* 3^k * Binomial[2*n+2,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, (k+1)*3^k*binomial(2*n+2, n-k));

Formula

a(n) = [x^n] 1/((1-4*x)^2 * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+2,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (k+1) * 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1)^2 ).
D-finite with recurrence 15*n*a(n) +2*(-94*n+23)*a(n-1) +192*(4*n-3)*a(n-2) +512*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Aug 21 2025

A386940 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 3, 13, 60, 285, 1378, 6748, 33372, 166365, 834900, 4213638, 21368724, 108820764, 556184580, 2851679620, 14661848560, 75568345821, 390330333402, 2020046912260, 10472193542100, 54373036935910, 282704274266040, 1471722678992700, 7670327017789800, 40017679829372700
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2025

Keywords

Crossrefs

Cf. A100193, A384365, A386941.
Cf. A000984, A293490.

Programs

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

Formula

a(n) = [x^n] 1/(sqrt(1-4*x) * (1-x)^n).
G.f.: (1+sqrt(1-4*x))/sqrt( 4 * (1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n-1/2,k) * binomial(n-k-1/2,n-k) = Sum_{k=0..n} (3/4)^k * binomial(2*k,k) * binomial(2*n-1/2,n-k).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n-1/2,k) * binomial(2*n-k-1,n-k).

A386942 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 8, 54, 340, 2060, 12180, 70812, 406656, 2313630, 13067340, 73372728, 410013864, 2282066332, 12658839200, 70017730680, 386314361808, 2126818591932, 11686657363236, 64108376373700, 351142219736000, 1920711937207140, 10493241496749000, 57263080117042800
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2025

Keywords

Links

Crossrefs

Cf. A293490, A377011, A383832.
Cf. A386940, A386941.

Programs

  • Magma
    [&+[(2*k+1) * Binomial (2*k, k) *Binomial(2*n-k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[(2*k+1) *Binomial[2*k,k]* Binomial[2*n-k,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*n-k, n-k));

Formula

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

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