A287584 Number of set partitions of [n] such that all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks are not larger than five.
1, 1, 2, 5, 15, 52, 203, 824, 3407, 14176, 58954, 244412, 1010802, 4167621, 17133558, 70278017, 287797888, 1177218237, 4811244031, 19651589669, 80234989720, 327503437323, 1336574600154, 5454075504109, 22254465906164, 90801509373219, 370472833209387
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Wikipedia, Partition of a set
Formula
G.f.: -(16*x^55 +40*x^54 +8*x^53 +74*x^52 -5*x^51 -318*x^50 -184*x^49 -329*x^48 -142*x^47 -724*x^46 -4295*x^45 +135*x^44 +10219*x^43 +11230*x^42 +17694*x^41 -9835*x^40 -58571*x^39 -44920*x^38 -18846*x^37 +77331*x^36 +137586*x^35 -2726*x^34 -66412*x^33 -120019*x^32 -106707*x^31 +110373*x^30 +61244*x^29 -89340*x^28 -166963*x^27 -241737*x^26 -18801*x^25 +183341*x^24 +76875*x^23 -44809*x^22 -194064*x^21 -276159*x^20 -117373*x^19 -3527*x^18 +50167*x^17 +68672*x^16 +6577*x^15 -20654*x^14 -18383*x^13 -13866*x^12 -2815*x^11 +2840*x^10 +1096*x^9 +484*x^8 +288*x^7 -290*x^6 -31*x^5 -4*x^4 -3*x^3 -2*x^2 +5*x -1) / ((x +1)*(x^5 -2*x^3 -2*x +1)*(16*x^40 +8*x^39 -8*x^38 -6*x^37 -25*x^36 -20*x^35 +20*x^34 -447*x^33 -222*x^32 -535*x^31 -399*x^30 +3024*x^29 +1695*x^28 +1438*x^27 +444*x^26 -7664*x^25 -2469*x^24 +1957*x^23 +290*x^22 -50*x^21 -6904*x^20 -7025*x^19 +2502*x^18 +2901*x^17 +352*x^16 -822*x^15 -8224*x^14 -7130*x^13 -2351*x^12 +680*x^11 +2679*x^10 +2620*x^9 +1264*x^8 +408*x^7 +62*x^6 -105*x^5 -48*x^4 -17*x^3 -5*x^2 -x +1)*(x -1)^2*(x^4 +x^3 +x^2 +x -1)^2).
a(n) = A000110(n) for n <= 6.