A287276 Number of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= four.

1, 1, 2, 5, 15, 52, 188, 696, 2606, 9800, 36931, 139303, 525658, 1983925, 7488281, 28265353, 106692425, 402731694, 1520195297, 5738304135, 21660476556, 81762200416, 308629323572, 1164989004846, 4397506361848, 16599351862867, 62657893108843, 236515956134402

Offset: 0

Author

Alois P. Heinz, May 22 2017

Keywords

• nonn
• easy

Examples

			a(6) = 188 = 203 - 15 = A000110(6) - 15 counts all set partitions of [6] except: 16|2345, 16|234|5, 16|235|4, 16|23|45, 16|23|4|5, 16|245|3, 16|24|35, 16|24|3|5, 16|25|34, 16|2|345, 16|2|34|5, 16|25|3|4, 16|2|35|4, 16|2|3|45, 16|2|3|4|5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
• Wikipedia, Partition of a set
• Index entries for linear recurrences with constant coefficients, signature (5,-4,-1,-7,7,0,1,-1).

Crossrefs

Column k=4 of A287214.

Cf. A000110.

Programs

Mathematica

LinearRecurrence[{5,-4,-1,-7,7,0,1,-1},{1,1,2,5,15,52,188,696},30] (* Harvey P. Dale, Jan 02 2021 *)

Formula

G.f.: -(x^7+x^5-6*x^4-x^2+4*x-1)/(x^8-x^7-7*x^5+7*x^4+x^3+4*x^2-5*x+1).

a(n) = A287214(n,4).

a(n) = A000110(n) for n <= 5.