A261724 a(n) is the number of ways of putting n labeled balls into 4 indistinguishable boxes such that each box contains at least 3 balls.
15400, 200200, 1611610, 10335325, 57962905, 297797500, 1439774336, 6662393738, 29844199346, 130445781284, 559533979466, 2365296391535, 9885290914059, 40944327590760, 168389163468240, 688631376550260, 2803570746766140, 11373212443859760, 46006062639998890
Offset: 12
Links
- Colin Barker, Table of n, a(n) for n = 12..1000
- I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 (third formula on page 16 is incorrect).
- Index entries for linear recurrences with constant coefficients, signature (30, -415, 3514, -20386, 85924, -272198, 661180, -1244717, 1822478, -2068955, 1802474, -1181760, 563888, -184752, 37152, -3456).
Programs
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Magma
[(1/12)*(-3^(n-2)*(n^2+5*n+18)+(1/64)*(2^(2*n+5)+3*2^n*(n^4+2*n^3+19*n^2+42*n+64)-16*(n^6-9*n^5+43*n^4-91*n^3+112*n^2-32*n+8))): n in [12..40]];
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Mathematica
Table[(1/12) (-(3^(n - 2) (n^2 + 5 n + 18)) + (1/64) (2^(2 n + 5) + 3 2^n (n^4 + 2 n^3 + 19 n^2 + 42 n + 64) - 16 (n^6 - 9 n^5 + 43 n^4 - 91 n^3 + 112 n^2 - 32 n + 8))), {n, 12, 40}] -
PARI
Vec(x^12*(15400 -261800*x +1996610*x^2 -9045575*x^3 +27162905*x^4 -57079715*x^5 +86268721*x^6 -94696602*x^7 +75062256*x^8 -41952000*x^9 +15705360*x^10 -3538080*x^11 +362880*x^12) / ((1 -x)^7*(1 -2*x)^5*(1 -3*x)^3*(1 -4*x)) + O(x^30)) \\ Colin Barker, May 24 2016
Formula
a(n) = (1/12)*(-3^(n - 2)*(n^2 + 5*n + 18) + (1/64)*(2^(2*n + 5) + 3*2^n*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64) - 16*(n^6 - 9*n^5 + 43*n^4 - 91*n^3 + 112*n^2 - 32*n + 8))).
G.f.: x^12*(15400 -261800*x +1996610*x^2 -9045575*x^3 +27162905*x^4 -57079715*x^5 +86268721*x^6 -94696602*x^7 +75062256*x^8 -41952000*x^9 +15705360*x^10 -3538080*x^11 +362880*x^12) / ((1 -x)^7*(1 -2*x)^5*(1 -3*x)^3*(1 -4*x)). - Colin Barker, May 24 2016
Extensions
Definition, data and formula corrected by Istvan Mezo and Bruno Berselli, May 20 2016
Comments