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A116364 Row squared sums of Catalan triangle A033184. 2
1, 2, 9, 60, 490, 4534, 45689, 489920, 5508000, 64276492, 773029466, 9531003552, 119990158054, 1537695160070, 20009930706137, 263883333450760, 3521003563829212, 47470845904561648, 645960472314074400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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, Sep 07 2007

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..830

FORMULA

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

EXAMPLE

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

MAPLE

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, Sep 07 2007

MATHEMATICA

Table[Sum[(Binomial[2*n-j+1, n-j]*(j+1)/(2*n-j+1))^2, {j, 0, n}], {n, 0, 30}] (* G. C. Greubel, May 12 2019 *)

PROG

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

(MAGMA) [(&+[(Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1))^2: j in [0..n]]): n in [0..30]]; // G. C. Greubel, May 12 2019

(Sage) [sum(( binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1) )^2 for j in (0..n)) for n in (0..30)] # G. C. Greubel, May 12 2019

(GAP) List([0..30], n-> Sum([0..n], j-> (Binomial(2*n-j+1, n-j)* (j+1)/(2*n-j+1))^2 )) # G. C. Greubel, May 12 2019

CROSSREFS

Cf. A033184, A116363.

Cf. A126217.

Sequence in context: A009636 A156272 A205570 * A120970 A111558 A322943

Adjacent sequences:  A116361 A116362 A116363 * A116365 A116366 A116367

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 04 2006

STATUS

approved

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Last modified July 20 22:20 EDT 2019. Contains 325189 sequences. (Running on oeis4.)