A058844 Number of ways of placing n labeled balls into 4 indistinguishable boxes with at least 2 balls in each box.

105, 1260, 9450, 56980, 302995, 1487200, 6914908, 30950920, 134779645, 575156036, 2417578670, 10046531276, 41388056231, 169371383384, 689568172832, 2796362035104, 11305163394129, 45595968007260, 183557935474290, 737897437077060, 2963015460969915

Offset

8,1

Links

T. D. Noe, Table of n, a(n) for n=8..200

Daniel J. Bernstein and Andreas Hülsing, Decisional second-preimage resistance: When does SPR imply PRE?, (2019).

Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.

Formula

E.g.f.: ((exp(x) - 1 - x)^4)/4!.

G.f.: x^8*(288*x^6 - 1560*x^5 + 3500*x^4 - 4130*x^3 + 2625*x^2 - 840*x + 105) / ((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2*(1 - 4*x)).

a(n) = (4^n-3^(n-1)(4n+12)+2^(n-1)(12+9n+3n^2)-4n^3-8n-4)/24. - David Wasserman, Jun 06 2007

Example

a(8) = 8!/(2!*2!*2!*2!*4!) = 105.

Maple

A058844:=n->(4^n-3^(n-1)*(4*n+12)+2^(n-1)*(12+9*n+3*n^2)-4*n^3-8*n-4)/24: seq(A058844(n), n=8..25); # Wesley Ivan Hurt, Oct 28 2014

Mathematica

Table[(4^n - 3^(n - 1) (4 n + 12) + 2^(n - 1) (12 + 9 n + 3 n^2) - 4 n^3 - 8 n - 4)/24, {n, 8, 25}] (* Wesley Ivan Hurt, Oct 28 2014 *)
offset = 8; terms = 21; egf = (Exp[x]-1-x)^4/4!; Drop[CoefficientList[egf + O[x]^(terms+offset), x]*Range[0, terms+offset-1]!, offset] (* Jean-François Alcover, May 07 2017 *)

Prog

(Magma) [(4^n-3^(n-1)*(4*n+12)+2^(n-1)*(12+9*n+3*n^2)-4*n^3-8*n-4)/24 : n in [8..25]]; // Wesley Ivan Hurt, Oct 28 2014
(PARI) a(n)=(4^n - 3^(n-1)*(4*n+12) + 2^(n-1)*(12+9*n+3*n^2) - 4*n^3 - 8*n - 4)/24 \\ Charles R Greathouse IV, Oct 28 2014

Crossrefs

Cf. A000247 (2 boxes), A000478 (3 boxes).

Sequence in context: A033593 A297542 A266105 * A297800 A165382 A051015

Adjacent sequences: A058841 A058842 A058843 * A058845 A058846 A058847

Keyword

easy,nonn

Author

Michael Steyer, Dec 02 2000

Extensions

More terms from James A. Sellers, Dec 06 2000

Status

approved