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

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

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

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.

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

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.

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

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.