cp's OEIS Frontend

A060488 Number of 4-block ordered tricoverings of an unlabeled n-set.

%I A060488 #43 Jun 12 2024 11:47:01
%S A060488 4,13,28,50,80,119,168,228,300,385,484,598,728,875,1040,1224,1428,
%T A060488 1653,1900,2170,2464,2783,3128,3500,3900,4329,4788,5278,5800,6355,
%U A060488 6944,7568,8228,8925,9660,10434,11248,12103,13000,13940,14924,15953,17028,18150,19320
%N A060488 Number of 4-block ordered tricoverings of an unlabeled n-set.
%C A060488 A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.
%C A060488 If Y is a 4-subset of an n-set X then, for n>=6, a(n-3) is the number of 3-subsets of X having at most one element in common with Y. - _Milan Janjic_, Dec 08 2007
%C A060488 Also the number of balls in a triangular pyramid of which all balls located on the edges have been removed such that the remaining pyramid's edges each consist of two adjacent balls. The layers of pyramids of this form start (from the top) 3, 7, 12, 18, 25, 33,... (A055998) with one smaller additional layer 1, 3, 6, 10, 15, 21,... (A000217) at the bottom. Thus, a(n) = A000217(n) + Sum_{k=1..n} A055998(k). Example: a(4) = (3+7+12+18)+10 = 50. - _K. G. Stier_, Dec 12 2012
%H A060488 Vincenzo Librandi, <a href="/A060488/b060488.txt">Table of n, a(n) for n = 3..1000</a>
%H A060488 Amya Luo, <a href="https://math.dartmouth.edu/theses/undergrad/2024/Luo-thesis.pdf">Pattern Avoidance in Nonnesting Permutations</a>, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
%H A060488 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A060488 a(n) = binomial(n+3, 3) - 6*binomial(n+1, 1) + 8*binomial(n, 0) - 3*binomial(n-1, -1).
%F A060488 G.f.: -y^3*(-4+3*y)/(-1+y)^4.
%F A060488 E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y)) * sum(k>=0, 1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k! ).
%F A060488 a(n) = (n+9)*binomial(n-1, 2)/3.
%F A060488 a(n) = (n-2)*(n-1)*(n+9)/6. - _Zak Seidov_, Jun 15 2006
%F A060488 a(3)=4, a(4)=13, a(5)=28, a(6)=50, a(n) = 4*a(n-1)-6*a(n-2)+ 4*a(n-3)- a(n-4). - _Harvey P. Dale_, Jul 21 2012
%t A060488 Table[ 3 (n - 1) (n - 2)/2! + n (n - 1) (n - 2)/3!, {n, 3, 62}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 14 2011 *)
%t A060488 Table[(n-2)(n-1)(n+9)/6,{n,3,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {4,13,28,50},50] (* _Harvey P. Dale_, Jul 21 2012 *)
%o A060488 (PARI) a(n)=(n-2)*(n-1)*(n+9)/6 \\ _Charles R Greathouse IV_, Jun 14 2011
%o A060488 (Magma) [(n-2)*(n-1)*(n+9)/6: n in [3..60]]; // _Vincenzo Librandi_, Jun 15 2011
%Y A060488 Essentially the same as A026054. - _Vladeta Jovovic_, Jun 15 2006
%Y A060488 Column k=4 of A060492.
%Y A060488 Cf. A060091, A060491.
%Y A060488 Fourth column (m=3) of (1, 4)-Pascal triangle A095666.
%K A060488 nonn,easy
%O A060488 3,1
%A A060488 _Vladeta Jovovic_, Mar 20 2001