A290597 Numerators in the expansion of the exponential generating function ((1 + 3*x)/x)*(1 - (1 + 3*x)^(-2/3)).

2, 1, -10, 20, -176, 6160, -29920, 523600, -96342400, 250490240, -6603833600, 581137356800, -6258402304000, 220832195584000, -25351536053043200, 348583620729344000, -15419698987556864000, 6553372069711667200000, -36560917862601932800000, 1945040830290422824960000, -327878311391814133350400000, 6468144870183969721548800000, -402149876711438117470208000000, 78620300897086151965425664000000, -1786253236381797372654471086080000, 127098787973320197669645058048000000

Offset: 0

Author

Wolfdieter Lang, Sep 13 2017

Keywords

• sign
• easy

Comments

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

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.

Examples

			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, ...}.
		

Links

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

Crossrefs

Cf. A038500, A290596, A290603 (z(3,2;n)).

Formula

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