%I #22 Aug 05 2024 14:02:38
%S 2,12,46,152,474,1444,4358,13104,39346,118076,354270,1062856,3188618,
%T 9565908,28697782,86093408,258280290,774840940,2324522894,6973568760,
%U 20920706362,62762119172,188286357606,564859072912,1694577218834,5083731656604,15251194969918
%N Sequence b_3 (n) arising from homology of partitions with even number of blocks.
%H Vincenzo Librandi, <a href="/A003993/b003993.txt">Table of n, a(n) for n = 2..1000</a>
%H S. Sundaram, <a href="https://citeseerx.ist.psu.edu/pdf/381cddb91c261a5123684d7b056557f0f8c66098">The homology of partitions with an even number of blocks</a>, J. Alg. Comb., 4 (1995), 69-92.
%H S. Sundaram, <a href="http://citeseerx.ist.psu.edu/pdf/08428845a2d7735bf0d5467a3b51ef55de495e3b">Plethysm, partitions with an even number of blocks and Euler numbers</a>, DIMACS Series, Vol. 24 (1996), 171-198, Amer. Math. Soc.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,3).
%F a(n) = 2*3^(n-1)-2*n. - _Vaclav Kotesovec_, Nov 19 2012
%F From _Colin Barker_, Jun 20 2019: (Start)
%F G.f.: 2*x^2*(1 + x) / ((1 - x)^2*(1 - 3*x)).
%F a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3) for n>4.
%F (End)
%p A003993 := proc(n) option remember; if n = 1 then 2 else 3*A003993(n-1)+4*n-2; fi; end;
%t Table[2 3^(n-1) - 2 n, {n, 2, 30}] (* _Vincenzo Librandi_, May 03 2013 *)
%o (Magma) [2*3^(n-1)-2*n: n in [2..30]]; // _Vincenzo Librandi_, May 03 2013
%o (PARI) Vec(2*x^2*(1 + x) / ((1 - x)^2*(1 - 3*x)) + O(x^40)) \\ _Colin Barker_, Jun 20 2019
%Y Cf. A003994.
%K nonn,easy
%O 2,1
%A Sheila Sundaram (sheila(AT)paris-gw.cs.miami.edu)