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%I #7 Sep 23 2017 04:56:32
%S 2,-1,14,-35,364,-14560,79040,-1521520,304304000,-852051200,
%T 24012352000,-2245154912000,25560225152000,-949379791360000,
%U 114305326879744000,-1643139073896320000,75777707878512640000,-33493746882302586880000,193911166160699187200000,-10684505255454525214720000,1862156630236360108851200000
%N Numerators in the expansion of the exponential generating function (1/2)*((1 + 3*x)/x)*(1 - (1 + 3*x)^(-4/3)).
%C The denominators are A038500(n+1), n >= 0.
%C This gives one half of the numerators of the z-sequence for the generalized unsigned Lah number Sheffer matrix Lah[3,2] = A290598.
%C 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.
%H Wolfdieter Lang, <a href="/A290597/a290597.log.txt">Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.</a>
%F 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)).
%F 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.
%e The rationals z(3,2;n) = 2*a(n)/A038500(n+1) begin:
%e {4, -2, 28/3, -70, 728, -29120/3, 158080, -3043040, 608608000/9, -1704102400, 48024704000, -4490309824000/3, ...}
%Y Cf. A007559, A038500, A290597 (z(3,1;n)), A290598.
%K sign,easy
%O 0,1
%A _Wolfdieter Lang_, Sep 13 2017