A232475 Number of preferential arrangements of n labeled elements when at least k=4 elements per rank are required.

1, 0, 0, 0, 1, 1, 1, 1, 71, 253, 673, 1585, 38149, 277707, 1402831, 5923503, 85577571, 937629969, 7475614341, 48939413477, 587610659505, 7906296686903, 87384175023995, 804959532778571, 9729015122635103, 144711323234918941, 2009073351016603121

Offset

0,9

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.

Formula

E.g.f.: 1/(2 + x - exp(x) + x^2/2 + x^3/6). - Vaclav Kotesovec, Aug 02 2014

a(n) ~ n! / ((1+r^3/6) * r^(n+1)), where r = 1.97615974210650519398... is the root of the equation 2 + r - exp(r) + r^2/2 + r^3/6 = 0. - Vaclav Kotesovec, Aug 02 2014

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

Maple

b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=4..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 29 2014

Mathematica

CoefficientList[Series[1/(2 + x - E^x + x^2/2 + x^3/6), {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Aug 02 2014 *)

Crossrefs

Cf. A032032, A102233.

Cf. column k=4 of A245732.

Sequence in context: A123038 A325079 A142325 * A243579 A142013 A033240

Adjacent sequences: A232472 A232473 A232474 * A232476 A232477 A232478

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2013

Extensions

More terms from Alois P. Heinz, Jul 29 2014

Status

approved