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A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262. 3
1, 1, 3, 7, 49, 321, 2851, 24823, 256257, 2887489, 36759331, 507010791, 7597222513, 122184356737, 2106356007939, 38693238713431, 754792977928321, 15572911248409473, 338800604611562947, 7749991799652960199, 185934065196259734321, 4667877395135551746241 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..444

FORMULA

E.g.f.: exp(x*(1-x^2+x^3)/(1-x)).

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 3*k, -1, -1]/k!, {k, 0, Floor[n/3]}], n=0, 1....

a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - Vaclav Kotesovec, Jun 22 2013

MAPLE

a:= proc(n) option remember; `if`(n=0, 1, add(

      `if`(j=3, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

MATHEMATICA

Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] - Zerinvary Lajos, Mar 23 2007

a[n_] := a[n] = If[n==0, 1, Sum[If[j==3, 0, a[n-j]*Binomial[n-1, j-1]*j!], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Feb 11 2017, after Alois P. Heinz *)

PROG

(PARI) x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ G. C. Greubel, May 17 2018

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(1-x^2+x^3)/(1-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

CROSSREFS

Cf. A052845, A113235.

Sequence in context: A190444 A118393 A113775 * A035499 A273092 A120788

Adjacent sequences:  A113233 A113234 A113235 * A113237 A113238 A113239

KEYWORD

nonn

AUTHOR

Karol A. Penson, Oct 19 2005

STATUS

approved

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Last modified September 19 04:25 EDT 2018. Contains 315155 sequences. (Running on oeis4.)