login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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
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.
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 A379843 A346378
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Sep 13 2017
STATUS
approved