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.

A358363 a(n) = 16^n * Sum_{k=0..n} (-1)^k*binomial(1/2, k)^2.

Original entry on oeis.org

1, 12, 196, 3120, 50020, 799536, 12799632, 204724416, 3276326820, 52413049520, 838703348496, 13418125153472, 214703825630736, 3435088134123200, 54963617747611200, 879389273444524800, 14070604335190692900, 225124668703739770800, 3602061930346132909200
Offset: 0

Views

Author

Peter Luschny, Nov 12 2022

Keywords

Crossrefs

Cf. A358362, A358364, A358365, A367331.

Programs

  • Maple
    a := n -> 16^n*add((-1)^k*binomial(1/2, k)^2, k = 0..n):
seq(a(n), n = 0..18);
  • Mathematica
    a[n_] := 16^n * Sum[(-1)^k*Binomial[1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)

Formula

a(n) = (64*(2*n - 3)^2*a(n - 2) + 12*(4*n - 3)*a(n - 1)) / n^2.
G.f.: hypergeom([-1/2, -1/2], [1], -16*x)/(1 - 16*x).
a(n) ~ sqrt(Pi) * 2^(4*n + 5/2) / Gamma(1/4)^2. - Vaclav Kotesovec, Nov 14 2023

A358364 a(n) = 16^n * Sum_{k=0..n} binomial(1/2, k)^2.

Original entry on oeis.org

1, 20, 324, 5200, 83300, 1333584, 21344400, 341580096, 5466017700, 87464462800, 1399525960976, 22393543798080, 358310523944464, 5733141459080000, 91732470946920000, 1467748145667974400, 23484346290765886500, 375754541311565499600, 6012139892071344570000
Offset: 0

Views

Author

Peter Luschny, Nov 12 2022

Keywords

Crossrefs

Cf. A358362, A358363, A358365, A367332.

Programs

  • Maple
    a := n -> 16^n*add(binomial(1/2, k)^2, k = 0..n):
seq(a(n), n = 0..18);
  • Mathematica
    a[n_] := 16^n * Sum[Binomial[1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)

Formula

a(n) = (16*n + 4)*(2*n - 1)^2*a(n - 1) / ((4*n - 3) * n^2).
G.f.: hypergeom([-1/2, -1/2], [1], 16*x)/(1 - 16*x).
a(n) ~ 2^(4*n+2) / Pi. - Vaclav Kotesovec, Nov 14 2023

A358365 a(n) = 16^n * Sum_{k=0..n} binomial(-1/2, k)^2.

Original entry on oeis.org

1, 20, 356, 6096, 102436, 1702480, 28093456, 461273920, 7546019620, 123100218320, 2003738272656, 32557446669120, 528231606378256, 8559878182412096, 138567392514153536, 2241139725237406976, 36219533239041063716, 584958249814679707856, 9441690077748181415696
Offset: 0

Views

Author

Peter Luschny, Nov 12 2022

Keywords

Crossrefs

Cf. A358362, A358363, A358364, A367333.

Programs

  • Maple
    a := n -> 16^n*add(binomial(-1/2, k)^2, k = 0..n):
seq(a(n), n = 0..18);
  • Mathematica
    a[n_] := 16^n * Sum[Binomial[-1/2, k]^2, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)

Formula

a(n) = (-64*(2*n - 1)^2*a(n - 2) + 4*(8*n^2 - 4*n + 1)*a(n - 1)) / n^2.
G.f.: hypergeom([1/2, 1/2], [1], 16*x)/(1 - 16*x).
G.f.: 2*EllipticK(4*sqrt(x))/(Pi*(1 - 16*x)).
a(n) ~ (log(n) + gamma + 4*log(2)) * 2^(4*n)/Pi, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 14 2023

A367330 a(n) = 27^n * Sum_{k=0..n} (-1)^k*binomial(-1/3, k)^2.

Original entry on oeis.org

1, 24, 684, 17880, 493785, 13108608, 358702272, 9579537792, 261039317220, 6992695897440, 190104989730480, 5101807912472160, 138496042650288420, 3721234160086727040, 100918032317551270080, 2713823288825315967360, 73545091414048811297745
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 14 2023

Keywords

Comments

In general, for m>1, Sum_{k>=0} (-1)^k * binomial(-1/m,k)^2 = 2^(-1/m) * sqrt(Pi) / (Gamma(1 - 1/(2*m)) * Gamma(1/2 + 1/(2*m))).

Crossrefs

Cf. A358362, A367331, A367332, A367333.

Programs

  • Mathematica
    Table[27^n*Sum[(-1)^k*Binomial[-1/3, k]^2, {k, 0, n}], {n, 0, 16}]

Formula

a(n) ~ Gamma(1/3)^3 * 3^(3*n+1) / (2^(8/3) * Pi^2).
Showing 1-4 of 4 results.

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