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A290603 Numerators in the expansion of the exponential generating function (1/2)*((1 + 3*x)/x)*(1 - (1 + 3*x)^(-4/3)). 2
2, -1, 14, -35, 364, -14560, 79040, -1521520, 304304000, -852051200, 24012352000, -2245154912000, 25560225152000, -949379791360000, 114305326879744000, -1643139073896320000, 75777707878512640000, -33493746882302586880000, 193911166160699187200000, -10684505255454525214720000, 1862156630236360108851200000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The denominators are A038500(n+1), n >= 0.

This gives one half of the numerators of the z-sequence for the generalized unsigned Lah number Sheffer matrix Lah[3,2] = A290598.

For Sheffer a- and z-sequences see a W. Lang link under A006232 with the references for the Riordan case, and also the present link for a proof.

LINKS

Table of n, a(n) for n=0..20.

Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

FORMULA

a(n) = numerator(r(n)) with the rationals r(n) = [x^n/n!] (1/2)*((1 + 3*x)/x)*(1 - (1 + 3*x)^(-4/3)).

2*a(n)/A038500(n+1) = z(3,2;n) = 4 for n = 0, and ((-1)^n/(n+1)*Product_{j=1..n} (1+3*j) = ((-1)^n/(n+1))*A007559(n+1) for n >= 1.

EXAMPLE

The rationals z(3,2;n) = 2*a(n)/A038500(n+1) begin:

{4, -2, 28/3, -70, 728, -29120/3, 158080, -3043040, 608608000/9, -1704102400, 48024704000, -4490309824000/3, ...}

CROSSREFS

Cf. A007559, A038500, A290597 (z(3,1;n)), A290598.

Sequence in context: A216445 A124026 A106204 * A083074 A181869 A141510

Adjacent sequences:  A290600 A290601 A290602 * A290604 A290605 A290606

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 13 2017

STATUS

approved

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Last modified August 22 17:42 EDT 2019. Contains 326180 sequences. (Running on oeis4.)