0,2

Number of 321-avoiding permutations in which the length of the longest increasing subsequence is n. Example: a(2)=9 because we have 12, 132, 312, 213, 231, 3142, 3412, 2143 and 2413. Column sums of triangle in A126217 (n>=1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 07 2007

a(n) = Sum_{k=0..n} [ C(2*n-k+1,n-k)*(k+1)/(2*n-k+1) ]^2.

The dot product of Catalan row 4 with itself equals

a(4) = [14,14,9,4,1]*[14,14,9,4,1] = 490

which is equivalent to obtaining the final term

in these repeated partial sums of Catalan row 4:

14,14, 9, 4, 1

..28,37,41,42

...65,106,148

....171,319

......490

a:=proc(k) options operator, arrow: sum((2*k-n+1)^2*binomial(n+1, k+1)^2/(n+1)^2, n=k..2*k) end proc: 1, seq(a(k), k=1..17); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 07 2007

(PARI) a(n)=sum(k=0, n, ((k+1)*binomial(2*n-k+1, n-k)/(2*n-k+1))^2)

Cf. A033184, A116363.

Cf. A126217.

Sequence in context: A151616 A009636 A156272 this_sequence A120970 A111558 A001193

Adjacent sequences: A116361 A116362 A116363 this_sequence A116365 A116366 A116367

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2006