A167892 a(n) = Sum_{k=1..n} Catalan(k)^2.

1, 5, 30, 226, 1990, 19414, 203455, 2248355, 25887399, 307993015, 3763786811, 47032778955, 598933188955, 7751562502555, 101741582076580, 1351906409905480, 18159677984049580, 246298405721739580, 3369517588450715680, 46457194476711692080

Offset: 1

Author

Alexander Adamchuk, Nov 15 2009

Keywords

• nonn

Comments

CatalanNumber(k) = (2k)!/(k!*(k+1)!) = binomial(2k,k)/(k+1).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
• Eric Weisstein's World of Mathematics, Catalan Number

Crossrefs

Cf. A000108, A014138, A167892, A167893, A001246, A033536, A014137, A094639.

Programs

Magma

[&+[Catalan(i)^2: i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Jul 01 2016

Mathematica

Array[n \[Function] Sum[CatalanNumber[k]^2, {k, 1, n}], 20] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
Accumulate[CatalanNumber[Range[1, 20]]^2] (* Vincenzo Librandi, Jul 01 2016 *)

Formula

a(n) = Sum_{k=1..n} Catalan(k)^2.

a(n) = Sum_{k=1..n} ((2k)!/(k!*(k+1)!))^2.

a(n) = Sum_{k=1..n} A000108(k)^2.

a(n) = Sum_{k=1..n} A001246(k).

a(n) = A094639(n) - 1.

G.f.: (Hypergeometric2F1(-1/2,-1/2,1,16*x) - 4*x - 1)/(4*x*(1 - x)). - Ilya Gutkovskiy, Jul 01 2016

Extensions

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010