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%I #10 Sep 15 2017 01:01:16
%S 2,1,-10,20,-176,6160,-29920,523600,-96342400,250490240,-6603833600,
%T 581137356800,-6258402304000,220832195584000,-25351536053043200,
%U 348583620729344000,-15419698987556864000,6553372069711667200000,-36560917862601932800000,1945040830290422824960000,-327878311391814133350400000,6468144870183969721548800000,-402149876711438117470208000000,78620300897086151965425664000000,-1786253236381797372654471086080000,127098787973320197669645058048000000
%N Numerators in the expansion of the exponential generating function ((1 + 3*x)/x)*(1 - (1 + 3*x)^(-2/3)).
%C The denominators are A038500(n+1), n >= 0.
%C The rationals z(n) = a(n)/A038500(n+1) give the Sheffer z-sequence for the generalized unsigned Lah triangle L[3,1] = A290596. 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.pdf">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 + 3*x)/x)*(1 - (1 + 3*x)^(-2/3)).
%e The rationals r(n) = z(3,1;n) = a(n)/A038500(n+1) begin: {2, 1, -10/3, 20, -176, 6160/3, -29920, 523600, -96342400/9, 250490240, -6603833600, 581137356800/3, -6258402304000, 220832195584000, -25351536053043200/3, 348583620729344000, ...}.
%Y Cf. A038500, A290596, A290603 (z(3,2;n)).
%K sign,easy
%O 0,1
%A _Wolfdieter Lang_, Sep 13 2017
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