A287277 Number of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= five.

1, 1, 2, 5, 15, 52, 203, 825, 3442, 14589, 62361, 267663, 1151563, 4960725, 21384434, 92216247, 397743421, 1715713298, 7401353547, 31929410019, 137745628418, 594249218505, 2563666285385, 11060009097685, 47714467256725, 205847216392033, 888055467635514

Offset: 0

Author

Alois P. Heinz, May 22 2017

Keywords

• nonn
• easy

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 (6,-6,-4,-3,-28,54,16,-16,6,-30,-4,13,0,2,0,-1).

Crossrefs

Column k=5 of A287214.

Cf. A000110.

Programs

Mathematica

CoefficientList[Series[-(x^15+x^14-x^13-12x^11-10x^10+17x^9+5x^8+20x^7+19x^6-31x^5- 4x^4- 3x^3-2x^2+5x-1)/((x^6+x^5-x^4-3x^2-x+1)(x^10-x^9-x^7-9x^6+10x^5+9x^4-7x^3+ 4x^2- 5x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-6,-4,-3,-28,54,16,-16,6,-30,-4,13,0,2,0,-1},{1,1,2,5,15,52,203,825,3442,14589,62361,267663,1151563,4960725,21384434,92216247},30] (* Harvey P. Dale, Jan 05 2024 *)

Formula

G.f.: -(x^15 +x^14 -x^13 -12*x^11 -10*x^10 +17*x^9 +5*x^8 +20*x^7 +19*x^6 -31*x^5 -4*x^4 -3*x^3 -2*x^2 +5*x-1) / ((x^6 +x^5 -x^4 -3*x^2 -x+1) * (x^10 -x^9 -x^7 -9*x^6 +10*x^5 +9*x^4 -7*x^3 +4*x^2 -5*x+1)).

a(n) = A287214(n,5).

a(n) = A000110(n) for n <= 6.