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

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

A367331 a(n) = 27^n * Sum_{k=0..n} (-1)^k*binomial(1/3, k)^2.

Original entry on oeis.org

1, 24, 657, 17664, 477828, 12888288, 348197220, 9397548288, 253804616001, 6851337236952, 185014241769825, 4994797849546752, 134872057740184128, 3641273395825798656, 98320397048549301312, 2654515896013953110016, 71674988018612154171876
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. A358363, A367330, A367332, A367333.

Programs

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

Formula

a(n) ~ 2^(5/3) * Pi * 3^(3*n + 1/2) / Gamma(1/3)^3.
Showing 1-4 of 4 results.

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