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.

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).

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.

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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