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.

A358362 a(n) = 16^n * Sum_{k=0..n} (-1)^k*binomial(-1/2, k)^2.

Original entry on oeis.org

1, 12, 228, 3248, 56868, 846384, 14395920, 218556096, 3662534436, 56236646576, 933921124752, 14445103689408, 238434118702864, 3706773418885824, 60917716297733184, 950622015752780544, 15571249887287040804, 243694280206569964464, 3981466564018425521424
Offset: 0

Views

Author

Peter Luschny, Nov 12 2022

Keywords

Crossrefs

Cf. A358363, A358364, A358365, A367330.

Programs

  • Maple
    a := n -> 16^n*add((-1)^k*binomial(-1/2, k)^2, k = 0..n):
seq(a(n), n = 0..19);
  • 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 - 1)^2*a(n - 2) + (16*n - 4)*a(n - 1)) / n^2.
G.f.: hypergeom([1/2, 1/2], [1], -16*x)/(16*x - 1).
G.f.: 2*EllipticK(4*I*sqrt(x))/(Pi*(1 - 16*x)).
a(n) ~ A014549 * 2^(4*n). - Vaclav Kotesovec, Nov 14 2023

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

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

A367332 a(n) = 27^n * Sum_{k=0..n} binomial(1/3, k)^2.

Original entry on oeis.org

1, 30, 819, 22188, 599976, 16212420, 437948784, 11828393820, 319437445365, 8626198419930, 232935493710231, 6289845008414760, 169838331029620344, 4585907100958922088, 123825507087143633976, 3343423515649756142760, 90275493748778836055964
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 14 2023

Keywords

Comments

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

Crossrefs

Cf. A358364, A367330, A367331, A367333.

Programs

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

Formula

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

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