A140982 If (a_n) is a sequence then let L(a_n)=(b_n) where b_n = a_n^2 - a_{n-1} a_{n+1}. The given sequence is the rows of the triangle obtained by computing L^2(binomial(n,k)).

1, 3, 1, 6, 16, 1, 10, 100, 50, 1, 15, 400, 750, 120, 1, 21, 1225, 6125, 3675, 245, 1, 28, 3136, 34300, 54880, 13720, 448, 1, 36, 7056, 148176, 518616, 345744, 42336, 756, 1, 45, 14400, 529200, 3556224, 5186160, 1693440, 113400, 1200, 1, 55, 27225, 1633500

Offset: 3

Author

Bruce E. Sagan, Jul 28 2008

Keywords

• nonn
• tabl

Comments

L(binomial(n,k)) gives the Narayana numbers, A001263.

Links

• Peter R. W. McNamara and Bruce E. Sagan, Infinite log-concavity: developments and conjectures, arXiv:0808.1065 [math.CO], 2008-2009.
• Peter R. W. McNamara and Bruce E. Sagan, Infinite log-concavity: developments and conjectures, Advances in Applied Mathematics, 44 (1) (2010), 1-15.

Crossrefs

Cf. A001263.

Programs

Mathematica

a[n_, k_] := 2 * Binomial[n, k]^2 * Binomial[n, k - 1] * Binomial[n, k - 2] / ((n - 1) n^2); Table[ a[n, k], {n, 2, 11}, {k, 2, n}] // Flatten (* Robert G. Wilson v, Aug 03 2008 *)

Formula

a(n,k) = binomial(n,k)^2 * binomial(n,k-1) * binomial(n,k-2) / (n*binomial(n,2)).

Extensions

More terms from Robert G. Wilson v, Aug 03 2008