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.

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).

A386941 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 7, 45, 276, 1645, 9618, 55468, 316620, 1792989, 10089420, 56482998, 314859636, 1748876220, 9684449908, 53487036420, 294732771280, 1620825793053, 8897604701130, 48766676365204, 266905699036900, 1458941915879910, 7965552023094600, 43444688665988700
Offset: 0

Views

Author

Seiichi Manyama, Aug 10 2025

Keywords

Crossrefs

Cf. A100193, A384365, A386940.

Programs

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

Formula

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

A386955 a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 7, 42, 235, 1262, 6594, 33780, 170475, 850230, 4200130, 20585228, 100220718, 485164988, 2337145360, 11210274408, 53567616267, 255110184486, 1211287208346, 5735765695260, 27093982041546, 127699233939684, 600650635811532, 2819989050992472, 13216897613555550
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A088218, A258431, A384365, A386956.

Programs

  • Magma
    [&+[(k+1) * 2^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 12 2025
  • Mathematica
    Table[Sum[(k+1)*2^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, (k+1)*2^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-3*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n-k-1,n-k).
G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (3*sqrt(1-4*x)-1)^2 ).
a(n) ~ n * 3^(2*n) / 2^(n+1). - Vaclav Kotesovec, Aug 12 2025

A386956 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 19, 282, 3763, 47294, 571950, 6733668, 77723187, 883589238, 9924844474, 110396411372, 1218075749934, 13348677037868, 145438914042172, 1576690043132376, 17018212213758771, 182983432175308710, 1960781840268630786, 20947171352106580284, 223169444039365834362
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A088218, A258431, A384365, A386955.
Cf. A386957.

Programs

  • Magma
    [&+[(k+1) * 8^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[(k+1) * 8^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, (k+1)*8^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k-1,n-k).
G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).
Showing 1-4 of 4 results.

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