A014316 Convolution of Catalan numbers and squares.

0, 1, 5, 15, 38, 93, 236, 641, 1869, 5779, 18663, 62179, 211909, 734651, 2581317, 9169247, 32867726, 118729269, 431756336, 1579232417, 5806059402, 21443620901, 79522304354, 295990940421, 1105391706392, 4140688363241, 15553731459744, 58573820469505

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) ~ 5 * 2^(2*n+2) / (27 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2018

G.f.: (1 + x)*(1 - sqrt(1 - 4*x))/(2*(1 - x)^3). - Ilya Gutkovskiy, Mar 21 2018

Maple

a:= proc(n) option remember; `if`(n<3, n*(3*n-1)/2,
      ((60*n^3-232*n^2+264*n-90) *a(n-1)
      -(90*n^3-308*n^2+257*n-24) *a(n-2)
      +2*(2*n-3)*(10*n^2-12*n+1) *a(n-3))/ (n*(10*n^2-32*n+23)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 10 2013

Mathematica

a[n_] := Sum[CatalanNumber[k]*(k-n)^2, {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 16 2017 *)

Prog

(PARI) a(n) = sum(i=0, n, i^2*(2*(n-i))!/((n-i)!*(n-i+1)!)); \\ Michel Marcus, Nov 10 2013

Crossrefs

Cf. A000108, A000290.

Sequence in context: A142964 A050488 A188282 * A075717 A062487 A084447

Adjacent sequences: A014313 A014314 A014315 * A014317 A014318 A014319

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved