A274868 Number of set partitions of [n] into exactly four blocks such that all odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

1, 2, 7, 14, 35, 70, 155, 310, 651, 1302, 2667, 5334, 10795, 21590, 43435, 86870, 174251, 348502, 698027, 1396054, 2794155, 5588310, 11180715, 22361430, 44731051, 89462102, 178940587, 357881174, 715795115, 1431590230, 2863245995, 5726491990, 11453115051

Offset: 4

Author

Alois P. Heinz, Jul 09 2016

Keywords

• nonn
• easy

Examples

			a(6) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46.
a(7) = 14: 137|24|5|6, 13|24|57|6, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..1000
• Pedro Fernando Fernández Espinosa and Agustín Moreno Cañadas, Homological Ideals as Integer Specializations of Some Brauer Configuration Algebras, arXiv:2104.00050 [math.RT], 2021.

Crossrefs

Column k=4 of A274537.

Programs

Mathematica

Drop[CoefficientList[Series[-x^4/((x - 1) (2 x - 1) (x + 1) (2 x^2 - 1)), {x, 0, 36}], x], 4] (* Michael De Vlieger, Jun 15 2021 *)

Formula

G.f.: -x^4/((x-1)*(2*x-1)*(x+1)*(2*x^2-1)).

From Ridouane Oudra, Jul 13 2023: (Start)

a(n) = x/6 + ((-1)^n - 1)*y, where x = 2^n - 3*sqrt(2)^n + 2 and y = (1/2)*sqrt(2)^(n-1) - (1/4)*sqrt(2)^n - 1/6.

a(n) = ((3 - (-1)^n)/12)*(4^floor(n/2) - 3*2^floor(n/2) + 2).

a(2n) = (4^n)/6 - 2^(n-1) + 1/3.

a(2n+1) = 2*a(2n). (End)