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.

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A067622 Consider the power series (x + 1)^(1/3) = 1 + x/3-x^2/9 + 5x^3/81 + ...; sequence gives numerators of coefficients.

Original entry on oeis.org

1, 1, -1, 5, -10, 22, -154, 374, -935, 21505, -55913, 147407, -1179256, 3174920, -8617640, 70664648, -194327782, 537259162, -13431479050, 37466757350, -104906920580, 884215473460, -2491879970660, 7042269482300, -59859290599550
Offset: 0

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Author

Benoit Cloitre, Feb 02 2002

Keywords

Comments

a(n) is also the numerator of the binomial coefficient C(k,n) evaluated at k=1/3, e.g. a(4) = (1/24)k(k-1)(k-2)(k-3), plug in k=1/3 and take numerator. - James R. Buddenhagen, Aug 16 2014

Crossrefs

Denominators are A067623.

Programs

  • Maple
    s := convert(taylor((x+1)^(1/3), x, 50), polynom): for n from 0 to 50 do printf(`%a,`, abs(numer(coeff(s, x, n)))) od;
seq(numer(subs(k=1/3,expand(binomial(k,n)))),n=0..50) # James R. Buddenhagen, Aug 16 2014

Formula

a(n) =(-1)^n*A004990(n)*A067623(n)/A000244(n); ignoring signs, a(n) =A038502(A004990(n)) =A038502(A034164(n-2)). a(n)'s sign is (-1)^(n+1) if n>0.

Extensions

Edited by Henry Bottomley and James Sellers, Feb 11 2002

A364658 Numerators of coefficients in expansion of (1 + x)^(2/3).

Original entry on oeis.org

1, 2, -1, 4, -7, 14, -91, 208, -494, 10868, -27170, 69160, -535990, 1401820, -3704810, 29638480, -79653415, 215532770, -5280552865, 14452039420, -39743108405, 329300041070, -913059204785, 2540686482880, -21278249294120, 59579098023536, -167279775219928, 12713262916714528
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 01 2023

Keywords

Examples

			(1 + x)^(2/3) = 1 + 2*x/3 - x^2/9 + 4*x^3/81 - 7*x^4/243 + 14*x^5/729 - 91*x^6/6561 + ...
Coefficients are 1, 2/3, -1/9, 4/81, -7/243, 14/729, -91/6561, ...

Crossrefs

Denominators are A067623.
Cf. A002596, A067622, A127974, A161200, A364661.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[(1 + x)^(2/3), {x, 0, nmax}], x] // Numerator
Table[Binomial[2/3, n], {n, 0, 27}] // Numerator
  • PARI
    my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(2/3))) \\ Michel Marcus, Aug 02 2023
Showing 1-2 of 2 results.

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