OFFSET
9,1
COMMENTS
a (k_1,n_1; k_2,n_2)-partition of a chain C is a chain of k_1+k_2 intervals of C, k_1 being of length n_1 and k_2 of length n_2.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 9..200 [a(67) and a(72) corrected by Georg Fischer, May 29 2021]
FORMULA
a(n) = (16/3)*(2*n-7)*(2*n-9)*(2*n-11)*(2*n-13)*(n-8)*(2*n-15)!/(n*(n-1)*(n-2)*(n-8)!^2).
D-finite with recurrence: (-9*n+n^2)*a(n) - (42-26*n+4*n^2)*a(n-1) = 0, a(9) = 220 for n >= 10. - Georg Fischer, May 29 2021
a(n) ~ n^(5/2) * 2^(2*n) / (384*sqrt(Pi)). - Vaclav Kotesovec, May 29 2021
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=9} 1/a(n) = sqrt(3)*Pi/20 - 5051/18900.
Sum_{n>=9} (-1)^(n+1)/a(n) = 65*sqrt(5)*log(phi)/6 - 15731/1350, where phi is the golden ratio (A001622). (End)
EXAMPLE
a(9)=220 because in the linearly ordered set {1,...,90} we can choose in 220 ways 12 successive blocks, 3 constituted of 3 consecutive elements and 9 of 9 consecutive elements.
MAPLE
rec:={(-9*n+n^2)*a(n) - (42-26*n+4*n^2)*a(n-1) = 0, a(9) = 220};
f:= gfun:-rectoproc(rec, a(n), remember): map(f, [$9..26]); # Georg Fischer, May 29 2021
PROG
(Magma) [16/3*(2*n-7)*(2*n-9)*(2*n-11)*(2*n-13)*(n-8)*Factorial(2*n-15)/(n*(n-1)*(n-2)*Factorial(n-8)^2): n in [9..30]]; // Vincenzo Librandi, Jun 30 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo Dominici (pl.dm(AT)libero.it), Jun 07 2000
STATUS
approved