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

0,3

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).

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.

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 *)

Crossrefs

Column k=5 of A287214.

Cf. A000110.

Sequence in context: A343666 A276722 A287584 * A287255 A284727 A056273

Adjacent sequences: A287274 A287275 A287276 * A287278 A287279 A287280

Keyword

nonn,easy

Author

Alois P. Heinz, May 22 2017

Status

approved