OFFSET
5,1
COMMENTS
Starting at n=7, the terms in the sequence alternate one odd and 3 even.
LINKS
Index entries for linear recurrences with constant coefficients, signature (12,-62,180,-321,360,-248,96,-16).
FORMULA
a(n) = Sum_{j=4..n/2+1} binomial(n,j)*binomial(j,3), n even.
a(n) = (Sum_{j=4..ceiling(n/2)} binomial(n,j)*binomial(j,3)) + (1/2)*binomial(n,ceiling(n/2)+1)*binomial(ceiling(n/2)+1,3), n odd.
From Alois P. Heinz, Nov 16 2021: (Start)
a(n) = binomial(n,3) * Stirling2(n-3,2).
G.f.: x^5*(8*x^6 - 48*x^5 + 124*x^4 - 180*x^3 + 145*x^2 - 60*x + 10)/((2*x-1)^4*(x-1)^4). (End)
E.g.f.: (1/12)*x^3*(exp(x)-1)^2.
a(n) = 12*a(n-1) - 62*a(n-2) + 180*a(n-3) - 321*a(n-4) + 360*a(n-5) - 248*a(n-6) + 96*a(n-7) - 16*a(n-8). - Wesley Ivan Hurt, Dec 03 2021
EXAMPLE
a(5)=10 since [5] can be written as the union of the following sets: {1,2,3,4} U {1,2,3,5}, {1,2,3,4} U {1,2,4,5}, {1,2,3,4} U {1,3,4,5}, {1,2,3,4} U {2,3,4,5}, {1,2,3,5} U {1,2,4,5}, {1,2,3,5} U {1,3,4,5},{1,2,3,5} U {2,3,4,5}, {1,2,4,5} U {1,3,4,5}, {1,2,4,5} U {2,3,4,5}, {1,3,4,5} U {2,3,4,5}.
MAPLE
a:= n-> binomial(n, 3)*Stirling2(n-3, 2):
seq(a(n), n=5..32); # Alois P. Heinz, Nov 16 2021
MATHEMATICA
nterms=50; Table[Binomial[n, 3]*StirlingS2[n-3, 2], {n, 5, nterms+4}] (* Paolo Xausa, Nov 20 2021 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Nov 16 2021
STATUS
approved