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

4,2

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.

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)

Example

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.

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 *)

Crossrefs

Column k=4 of A274537.

Sequence in context: A227213 A319455 A060552 * A191396 A173126 A256272

Adjacent sequences: A274865 A274866 A274867 * A274869 A274870 A274871

Keyword

nonn,easy

Author

Alois P. Heinz, Jul 09 2016

Status

approved