A285059 Numerators of the exponential expansion of (4/(3*log(1+x)))*(1 - 1/(1+x)^(3/4)).

1, -3, 9, -363, 6411, -46569, 3615627, -108267435, 2044658079, -27994845375, 5887932942123, -90460390681593, 475997756735954241, -3681053425472669991, 14270353890553782297, -2661381204559253577387, 880641541680797362210263

Offset

0,2

Comments

For the denominators see A285060.

This gives one third of the numerators of the z-sequence for the Sheffer triangle (exp(3*x), exp(4*x) - 1) shown in A225467. For the notion and use of a- and z- sequences for Sheffer triangles see the W. Lang link under A006232, also for references. The a-sequence of this Sheffer triangle is given by 4*A006232/A006233.

For the nontrivial recurrence of {3^n} given by the z-sequence for the m = 0 column of the triangle A225467 see the example for n = 3 below.

Links

Table of n, a(n) for n=0..16.

Formula

E.g.f.: (4/(3*log(1+x)))*(1 - 1/(1+x)^(3/4)) for the rational sequence a(n)/A285060(n), n >= 0.

Example

The rationals a(n)/A285060(n) start: 1, -3/8, 9/16, -363/256, 6411/1280, -46569/2048, 3615627/28672, -108267435/131072, 2044658079/327680, -27994845375/524288, ...
From the z-recurrence for A225467(3, 0) = 3^3 = 27 one finds: 3^3 = 3*3*(1*9 + 40*(-3/8) + 16*(9/16)).

Crossrefs

Cf. A006232/A006233 (a-sequence), A284857/A284858 (case [3,1]), A225467.

Sequence in context: A132562 A303130 A144984 * A260551 A359615 A137036

Adjacent sequences: A285056 A285057 A285058 * A285060 A285061 A285062

Keyword

sign,frac,easy

Author

Wolfdieter Lang, Apr 13 2017

Status

approved