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 1-7 of 7 results.

A331656 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^k.

Original entry on oeis.org

1, 3, 37, 847, 28401, 1256651, 69125869, 4548342975, 348434664769, 30463322582899, 2993348092318101, 326572612514776079, 39170287549040392369, 5123157953193993402171, 725662909285939100555101, 110662236267661479984580351, 18077209893508013563092846849
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2020

Keywords

Links

Crossrefs

Main diagonal of A335333.
Cf. A001850, A006442, A084768, A084769, A110129, A331657.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^k, {k, 0, n}], {n, 1, 16}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (2 n + 1) x + x^2], {x, 0, n}], {n, 0, 16}]
Table[LegendreP[n, 2 n + 1], {n, 0, 16}]
Table[Hypergeometric2F1[-n, n + 1, 1, -n], {n, 0, 16}]
  • PARI
    a(n) = {sum(k=0, n, binomial(n,k) * binomial(n+k,k) * n^k)} \\ Andrew Howroyd, Jan 23 2020

Formula

a(n) = central coefficient of (1 + (2*n + 1)*x + n*(n + 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 - 2*(2*n + 1)*x + x^2).
a(n) = n! * [x^n] exp((2*n + 1)*x) * BesselI(0,2*sqrt(n*(n + 1))*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * (n + 1)^(n - k).
a(n) = P_n(2*n+1), where P_n is n-th Legendre polynomial.
a(n) ~ exp(1/2) * 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Jan 28 2020
From Seiichi Manyama, Aug 30 2025: (Start)
a(n) = (-1)^n * Sum_{k=0..n} (1/(2*(2*n+1)))^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k).
a(n) = Sum_{k=0..floor(n/2)} (n*(n+1))^k * (2*n+1)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)

A331657 a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n,k) * binomial(n+k,k) * n^k.

Original entry on oeis.org

1, 1, 13, 305, 10321, 458649, 25289461, 1666406209, 127779121345, 11178899075537, 1098961472475901, 119937806278590321, 14389588419704763409, 1882432013890951832425, 266678501426944160023653, 40673387011956179149166849, 6644919093900517186643470081
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2020

Keywords

Links

Crossrefs

Cf. A001850, A006442, A084768, A084769, A110129, A331656, A335333.

Programs

  • Magma
    [&+[(-1)^(n-k)*Binomial(n,k)*Binomial(n+k,k)*n^k:k in [0..n]]:n in [0..16]]; // Marius A. Burtea, Jan 23 2020
  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[n + k, k] n^k, {k, 0, n}], {n, 1, 16}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (2 n - 1) x + x^2], {x, 0, n}], {n, 0, 16}]
Table[LegendreP[n, 2 n - 1], {n, 0, 16}]
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 1, n], {n, 0, 16}]
  • PARI
    a(n) = {sum(k=0, n, (-1)^(n - k) * binomial(n,k) * binomial(n+k,k) * n^k)} \\ Andrew Howroyd, Jan 23 2020

Formula

a(n) = central coefficient of (1 + (2*n - 1)*x + n*(n - 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 - 2*(2*n - 1)*x + x^2).
a(n) = n! * [x^n] exp((2*n - 1)*x) * BesselI(0,2*sqrt(n*(n - 1))*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k * (n - 1)^(n - k).
a(n) = P_n(2*n-1), where P_n is n-th Legendre polynomial.
a(n) = (-1)^n * 2F1(-n, n + 1; 1; n).
a(n) ~ 4^n * n^(n - 1/2) / (exp(1/2) * sqrt(Pi)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 30 2025: (Start)
a(n) = (-1)^n * Sum_{k=0..n} (1/(2*(2*n-1)))^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k).
a(n) = Sum_{k=0..floor(n/2)} ((n-1)*n)^k * (2*n-1)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)

A349077 a(n) = 4^n * P(2*n, n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 4, 886, 575296, 748553926, 1638884021248, 5430931463592636, 25386301852394340352, 159203574262026117932614, 1290247693627696897075707904, 13126820230906199855332092508756, 163819123650250694146607819756929024, 2460884002303138397686849151579559249436
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A008316, A110129.

Programs

  • Mathematica
    Table[4^n*LegendreP[2*n, n], {n, 0, 15}]
  • PARI
    a(n) = 4^n*pollegendre(2*n, n); \\ Michel Marcus, Nov 08 2021

Formula

a(n) ~ 2^(4*n - 1/2) * n^(2*n - 1/2) / sqrt(Pi).

A349115 a(n) = 8^n * P(n, 3*n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 24, 3424, 926208, 369378816, 194988441600, 128184980586496, 100904418485993472, 92542260511611682816, 96909547417109671182336, 114095278582299648325582848, 149184455262733048487847395328, 214496285274348399077675463868416, 336346643957900669242934177071890432
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2021

Keywords

Comments

In general, for k>=1, P(n, k*n) ~ 2^n * k^n * n^(n - 1/2) / sqrt(Pi).

Links

Crossrefs

Cf. A008316, A110129, A349113, A349114.

Programs

  • Mathematica
    Table[8^n*LegendreP[n, 3*n], {n, 0, 15}]
  • PARI
    a(n) = 8^n*pollegendre(n, 3*n); \\ Michel Marcus, Nov 08 2021

Formula

a(n) ~ 2^(4*n) * 3^n * n^(n - 1/2) / sqrt(Pi).

A387430 a(n) = Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

Original entry on oeis.org

1, 2, 26, 576, 18886, 822800, 44758244, 2920443904, 222277449286, 19333107926208, 1891679562586252, 205658657276205056, 24594577004735218716, 3208651043895419972096, 453493188773477070618248, 69025100503218462336614400, 11256667883184684951198851654, 1958143582960886584057480612864
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2025

Keywords

Comments

Sum_{k=0..n} (n-i)^k * (n+i)^(n-k) * binomial(n,k) = 2^n * n^n. - Vaclav Kotesovec, Aug 29 2025

Links

Crossrefs

Main diagonal of A386621.
Cf. A110129, A387459.

Programs

  • Magma
    [&+[n^(n-2*k) *Binomial(2*(n-k),n-k) * Binomial(n-k,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
  • Mathematica
    Join[{1}, Table[Sum[(n^2 + 1)^k * (2*n)^(n-2*k) * Binomial[n,2*k] * Binomial[2*k,k], {k,0,n/2}], {n,1,20}]] (* or *)
Table[(I + n)^n Hypergeometric2F1[-n, -n, 1, (-I + n)/(I + n)], {n, 0, 20}] (* Vaclav Kotesovec, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, (n^2+1)^k*(2*n)^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} n^(n-2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).
a(n) = Sum_{k=0..floor(n/2)} (n^2+1)^k * (2*n)^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1 + 2*n*x + (n^2+1)*x^2)^n.
a(n) ~ 2^(2*n) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 29 2025

A349113 a(n) = 8^n * P(3*n, n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 8, 40636, 748832256, 37759888297756, 4086692369433395200, 815254385427670754825764, 270587150855247020644760551424, 138859707622050969870951620062449436, 104286590422721059977069662227099300134912, 109828573459404650800550127862919905133973562480
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2021

Keywords

Comments

In general, for k>=1, P(k*n, n) ~ 2^(k*n) * n^(k*n) / sqrt(k*Pi*n).

Links

Crossrefs

Cf. A008316, A110129, A349077, A349115.

Programs

  • Mathematica
    Table[8^n*LegendreP[3*n, n], {n, 0, 12}]
  • PARI
    a(n) = 8^n*pollegendre(3*n, n); \\ Michel Marcus, Nov 08 2021

Formula

a(n) ~ 2^(6*n) * n^(3*n - 1/2) / sqrt(3*Pi).

A349114 a(n) = 4^n * P(n, 2*n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 8, 376, 33984, 4526176, 797459200, 174910868224, 45926958135296, 14047764722238976, 4905641267399503872, 1925859774286175997952, 839619968812285810868224, 402496047174560754869846016, 210424519428145503482634174464, 119148510992477432889126160826368
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2021

Keywords

Links

Crossrefs

Cf. A008316, A110129, A349077, A349115.

Programs

  • Mathematica
    Table[4^n*LegendreP[n, 2*n], {n, 0, 16}]
  • PARI
    a(n) = 4^n*pollegendre(n, 2*n); \\ Michel Marcus, Nov 08 2021

Formula

a(n) ~ 2^(4*n) * n^(n - 1/2) / sqrt(Pi).
Showing 1-7 of 7 results.

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