A000478 Number of ways of placing n labeled balls into 3 indistinguishable boxes with at least 2 balls in each box. (Formerly M4978 N2138)

15, 105, 490, 1918, 6825, 22935, 74316, 235092, 731731, 2252341, 6879678, 20900922, 63259533, 190957923, 575363776, 1731333808, 5205011031, 15638101281, 46962537810, 140988276150, 423174543025, 1269959836015, 3810785476980, 11434235478348, 34306598748315, 102927849307725

Offset

6,1

Comments

Associated Stirling numbers.

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.

F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=6..200

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

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.

Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

Formula

E.g.f.: ((exp(x) - 1 - x)^3)/3!.

G.f.: x^6*(12*x^3 - 40*x^2 + 45*x - 15)/((1 - x)^3*(1 - 2*x)^2*(3*x - 1)). - Simon Plouffe in his 1992 dissertation

a(n) = (1+n+n^2)/2 - (1/2 + n/4)*2^n + 3^n/6. - Michael Steyer (m.steyer(AT)osram.de), Jan 09 2005

a(n) = 10*a(n-1) - 40*a(n-2) + 82*a(n-3) - 91*a(n-4) + 52*a(n-5) - 12*a(n-6), n > 11. - Harvey P. Dale based on Michael Steyer's formula, Jul 17 2011

a(n) = 3*a(n-1) + (2^(n-3)-n+1)*(n-1), a(n)=0, n < 6. - Vladimir Kruchinin, Oct 04 2018

Example

a(6) = 6!/(2!*2!*2!*3!) = 15.

Mathematica

Table[(1+n+n^2)/2-(1/2+n/4)*2^n+3^n/6, {n, 6, 30}] (* or *) LinearRecurrence[ {10, -40, 82, -91, 52, -12}, {15, 105, 490, 1918, 6825, 22935}, 25] (* Harvey P. Dale, Jul 17 2011 *)
offset = 6; terms = 26; egf = (Exp[x]-1-x)^3/3!; Drop[CoefficientList[egf + O[x]^(terms+offset), x]*Range[0, terms+offset-1]!, offset] (* Jean-François Alcover, May 07 2017 *)

Prog

(PARI) Vec(x^6*(12*x^3-40*x^2+45*x-15)/((1-x)^3*(1-2*x)^2*(3*x-1))+O(x^66)) /* Joerg Arndt, Apr 10 2013 */
(Python) # based on Vladimir Kruchinin's formula
def A000478():
    a = 15; n = 7; z = 4; s = 15;
    while True:
        yield a
        z = 2*z; s += n*(z-2) + 3; a = 3*a + s; n += 1
a = A000478(); print([next(a) for _ in range(6, 32)]) # Peter Luschny, Oct 04 2018

Crossrefs

Cf. A000247 (2 boxes), A058844 (4 boxes).

Sequence in context: A076767 A022610 A006857 * A055848 A202493 A200852

Adjacent sequences: A000475 A000476 A000477 * A000479 A000480 A000481

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Additional comments from Michael Steyer, Dec 02 2000

More terms from James A. Sellers, Dec 06 2000

More terms from Joerg Arndt, Apr 10 2013

Status

approved