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.

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

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.

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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