A047163 Number of nonempty subsets of {1,2,...,n} in which exactly 1/4 of the elements are <= n/2.

0, 0, 0, 0, 2, 3, 12, 16, 40, 50, 110, 135, 315, 392, 980, 1232, 3080, 3864, 9480, 11850, 29040, 36300, 89870, 112519, 280423, 351351, 876603, 1098188, 2741102, 3433703, 8586788, 10758608, 26965808, 33794504, 84844280, 106344792, 267298650, 335061789, 843098172, 1056924666

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

a(n) = Sum_{k>=1} binomial(floor(n/2), k)*binomial(ceiling(n/2), 3*k). - Andrew Howroyd, Apr 11 2021

Example

The a(7) = 12 qualifying subsets of {1,2,3,4,5,6,7} all have 4 elements with 1 element less than 7/2, and these subsets are {m,4,5,6}, {m,4,5,7}, {m,4,6,7}, {m,5,6,7} for 1 <= m <= 3.

Prog

(PARI) a(n) = {my(m=n\2); sum(k=1, (n-m)\3, binomial(m, k)*binomial(n-m, 3*k))} \\ Andrew Howroyd, Apr 11 2021

Crossrefs

Cf. A047161, A047164.

Sequence in context: A345362 A392181 A333319 * A046486 A073452 A112976

Adjacent sequences: A047160 A047161 A047162 * A047164 A047165 A047166

Keyword

nonn

Author

Clark Kimberling

Extensions

Terms a(35) and beyond from Andrew Howroyd, Apr 11 2021

Status

approved