A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

0, 1, 14, 75, 265, 735, 1736, 3654, 7050, 12705, 21670, 35321, 55419, 84175, 124320, 179180, 252756, 349809, 475950, 637735, 842765, 1099791, 1418824, 1811250, 2289950, 2869425, 3565926, 4397589, 5384575, 6549215, 7916160, 9512536

Offset: 2

Author

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

Keywords

• nonn

Comments

All terms are positive: A093566 >= A054563 ==> C( C(n,2), 3) >= C( C(n,3), 2) ==> n^2*(n^4 + 3n^3 -35n^2 + 69n -38)/144 >= 0 ==> (n - 2)(n - 1)(n^2 + 6n - 19) ==> 0 which it is for all n >= 2.

Links

• Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A054563, A093566.

Programs

Maple

A093567:=n->binomial(binomial(n, 2), 3) - binomial(binomial(n, 3), 2); seq(A093567(n), n=2..30); # Wesley Ivan Hurt, Feb 02 2014

Mathematica

Table[ Binomial[ Binomial[n, 2], 3] - Binomial[ Binomial[n, 3], 2], {n, 2, 34}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,14,75,265,735,1736},40] (* Harvey P. Dale, Jun 12 2016 *)

PARI

a(n) = binomial(binomial(n,2), 3) - binomial(binomial(n,3), 2); \\ Michel Marcus, Oct 01 2017

Formula

a(n) = A093566(n) - A054563(n).

G.f.: x^3*(-1-7*x+2*x^2+x^3)/(x-1)^7. - R. J. Mathar, Dec 08 2010