A290603 Numerators in the expansion of the exponential generating function (1/2)*((1 + 3*x)/x)*(1 - (1 + 3*x)^(-4/3)).

2, -1, 14, -35, 364, -14560, 79040, -1521520, 304304000, -852051200, 24012352000, -2245154912000, 25560225152000, -949379791360000, 114305326879744000, -1643139073896320000, 75777707878512640000, -33493746882302586880000, 193911166160699187200000, -10684505255454525214720000, 1862156630236360108851200000

Offset: 0

Author

Wolfdieter Lang, Sep 13 2017

Keywords

• sign
• easy

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.

Examples

			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, ...}
		

Links

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

Crossrefs

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

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.