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.

A386957 a(n) = Sum_{k=0..n} 8^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 11, 114, 1163, 11806, 119646, 1211820, 12271179, 124251318, 1258065866, 12737997724, 128972535582, 1305848105836, 13221716621852, 133869898347264, 1355432788629963, 13723757247851046, 138953043155444562, 1406899565919247884, 14244858120395937738, 144229188529316725956
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A000302, A000984, A026641, A141223, A377011.
Cf. A386958, A386960.

Programs

  • Magma
    [&+[8^k * Binomial(2*n+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 2025
  • Mathematica
    Table[Sum[8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 14 2025 *)
  • PARI
    a(n) = sum(k=0, n, 8^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-9*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(2*n-k,n-k).
G.f.: 2/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7) ).
D-finite with recurrence 8*n*a(n) +(-113*n+16)*a(n-1) +162*(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

A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 0, 4, 8, 36, 120, 456, 1680, 6340, 23960, 91224, 348656, 1337896, 5149872, 19877904, 76907808, 298176516, 1158168792, 4505865144, 17555689008, 68490100536, 267518448912, 1046041377264, 4094231982048, 16039426479336, 62887835652720, 246761907761776, 968943740083040
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Links

Crossrefs

Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.
Cf. A226733, A226705, A226751.
Cf. A005810, A079589, A183160.

Programs

  • Magma
    [&+[(-3)^(n-k) * Binomial(2*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[(-3)^(n-k)*Binomial[2*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));

Formula

a(n) = [x^n] (1+x)^(2*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n-k,n-k).
G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 19 2025

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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