A272982 a(n) is the number of ways of putting n labeled balls into 3 indistinguishable boxes such that each box contains at least 3 balls.

280, 2100, 10395, 42735, 158301, 549549, 1827826, 5903898, 18682014, 58257810, 179765973, 550478241, 1676305723, 5083927299, 15372843684, 46383762084, 139730030100, 420448298400, 1264071094975, 3798101973315, 11406989362185, 34248214131465, 102803026929030, 308533903071390

Offset

9,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 9..1000

I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 (second formula on page 16 is incorrect).

Index entries for linear recurrences with constant coefficients, signature (14,-85,294,-639,906,-839,490,-164,24).

Formula

G.f.: x^9*(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5).

a(n) = (1/3)*(1/16)*(6*n^4 - 12*n^3 - 3*2^n*n^2 + 42*n^2 - 9*2^n*n + 12*n + 8*3^n - 3*2^(n+3) + 24).

a(n) = 3*a(n-1) + C(n-1,2)*(2^(n-4) + 2 - n - C(n-3, 2)), a(n)=0, n < 9. - Vladimir Kruchinin, Oct 04 2018

Example

For n=9, label the balls A through I. The box containing ball A can contain 8*7/2 = 28 combinations of other balls. There are 6 balls for the other two boxes, so there are A272352(6) = 10 combinations for those two boxes. Thus, a(9) = 28*10 = 280. - Michael B. Porter, Jul 01 2016

Mathematica

Table[(1/3) (1/16) (6 n^4 - 12 n^3 - 3 2^n n^2 + 42 n^2 - 9 2^n n + 12 n + 8 3^n - 3 2^(n + 3) + 24), {n, 9, 40}]
CoefficientList[Series[(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5), {x, 0, 40}], x] (* Stefano Spezia, Oct 04 2018 *)

Prog

(Magma) [(1/3)*(1/16)*(6*n^4-12*n^3-3*2^n*n^2+42*n^2-9*2^n*n+12*n+8*3^n-3*2^(n+3)+24): n in [9..40]];
(PARI) Vec(x^9*(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5) + O(x^40)) \\ Stefano Spezia, Oct 04 2018

Crossrefs

Cf. A000478, A058844, A261724, A272352, column 3 of A059022.

Sequence in context: A187178 A233724 A233718 * A234458 A234452 A297510

Adjacent sequences: A272979 A272980 A272981 * A272983 A272984 A272985

Keyword

nonn,easy

Author

Vincenzo Librandi, May 12 2016

Extensions

Data, formulas and programs corrected for erroneous formula in Mezo's paper by Bruno Berselli, May 21 2016

Status

approved