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

%I A058844 #29 Jul 02 2025 16:02:00
%S A058844 105,1260,9450,56980,302995,1487200,6914908,30950920,134779645,
%T A058844 575156036,2417578670,10046531276,41388056231,169371383384,
%U A058844 689568172832,2796362035104,11305163394129,45595968007260,183557935474290,737897437077060,2963015460969915
%N A058844 Number of ways of placing n labeled balls into 4 indistinguishable boxes with at least 2 balls in each box.
%H A058844 T. D. Noe, <a href="/A058844/b058844.txt">Table of n, a(n) for n=8..200</a>
%H A058844 Daniel J. Bernstein and Andreas Hülsing, <a href="https://sphincs.org/data/dspr-20190513.pdf">Decisional second-preimage resistance: When does SPR imply PRE?</a>, (2019).
%H A058844 Erik Vigren and Andreas Dieckmann, <a href="https://doi.org/10.3390/sym14061090">A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks</a>, Symmetry (2022) Vol. 14, No. 6, 1090.
%F A058844 E.g.f.: ((exp(x) - 1 - x)^4)/4!.
%F A058844 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)).
%F A058844 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
%e A058844 a(8) = 8!/(2!*2!*2!*2!*4!) = 105.
%p A058844 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
%t A058844 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 *)
%t A058844 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 *)
%o A058844 (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
%o A058844 (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
%Y A058844 Cf. A000247 (2 boxes), A000478 (3 boxes).
%K A058844 easy,nonn
%O A058844 8,1
%A A058844 _Michael Steyer_, Dec 02 2000
%E A058844 More terms from _James Sellers_, Dec 06 2000