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

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

A367333 a(n) = 27^n * Sum_{k=0..n} binomial(-1/3, k)^2.

Original entry on oeis.org

1, 30, 846, 23430, 643635, 17601732, 480016620, 13065872292, 355170348720, 9644965082940, 261716257738980, 7097365769203260, 192376104782028120, 5212313820585819540, 141177183151026767580, 3822747528826291049460, 103486045894075138514445
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 14 2023

Keywords

Comments

Compare with A358365: Sum_{k>=0} binomial(-1/3, k)^2 converges, but Sum_{k>=0} binomial(-1/2, k)^2 diverges.
In general, for m>2, Sum_{k>=0} binomial(-1/m,k)^2 = Gamma(1 - 2/m) / Gamma(1 - 1/m)^2.

Crossrefs

Cf. A358365, A367330, A367331, A367332.

Programs

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

Formula

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

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