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.

A386837 a(n) = Sum_{k=0..n} binomial(4*n+2,k) * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 9, 126, 1978, 32703, 556887, 9665476, 170006256, 3019802253, 54047520709, 973141183002, 17607177876438, 319855973830251, 5830329608105763, 106583422441886592, 1953315343946213804, 35875864591309216089, 660185366847433991025, 12169379986275311820790
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386835, A386836.
Cf. A370101, A386834.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*n+2, k)*binomial(4*n-k-1, n-k));

Formula

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

A386833 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 6, 59, 656, 7701, 93210, 1150495, 14395428, 181936169, 2317140014, 29691138099, 382334271544, 4943464235069, 64137141682242, 834561532624967, 10886878474010700, 142332442919829585, 1864423992564121686, 24464149489904517211, 321499324010641490016, 4230840338116037836901
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A116881, A386834.
Cf. A370097, A386836.
Cf. A383326.

Programs

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

Formula

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

A386835 a(n) = Sum_{k=0..n} binomial(2*n+2,k) * binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 5, 30, 198, 1375, 9843, 71876, 532220, 3981645, 30023265, 227803642, 1737227682, 13303481035, 102234258623, 787997000640, 6089345072056, 47161769198809, 365986358229645, 2845097133606422, 22151577531840830, 172710278146819959, 1348274852150114251
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386836, A386837.
Cf. A116881.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n + 2, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(2*n-k-1, n-k));

Formula

a(n) = [x^n] (1+x)^(2*n+2)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^3 * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(n-k+2,n-k).
a(n) ~ 2^(3*n+5) / (27*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 06 2025

A386866 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+2,k) * binomial(3*n-k-1,n-k).

Original entry on oeis.org

1, 9, 132, 2197, 38649, 701292, 12979360, 243541725, 4616122851, 88173726337, 1694554311888, 32728267058604, 634701136059532, 12351249029265816, 241061116082196072, 4716751239386395885, 92494719333403946583, 1817328001770278062299, 35768122814759119268788
Offset: 0

Views

Author

Seiichi Manyama, Aug 06 2025

Keywords

Crossrefs

Cf. A384950, A386863.
Cf. A386836.

Programs

  • Mathematica
    Table[Sum[2^(n-k) Binomial[3n+2,k]Binomial[3n-k-1,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 02 2025 *)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+2, k)*binomial(3*n-k-1, n-k));

Formula

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

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