A167892 - OEIS

%I #32 Sep 08 2022 08:45:48

%S 1,5,30,226,1990,19414,203455,2248355,25887399,307993015,3763786811,

%T 47032778955,598933188955,7751562502555,101741582076580,

%U 1351906409905480,18159677984049580,246298405721739580,3369517588450715680,46457194476711692080

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

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

%H G. C. Greubel, <a href="/A167892/b167892.txt">Table of n, a(n) for n = 1..500</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>

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

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

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

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

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

%F G.f.: (Hypergeometric2F1(-1/2,-1/2,1,16*x) - 4*x - 1)/(4*x*(1 - x)). - _Ilya Gutkovskiy_, Jul 01 2016

%t Array[n \[Function] Sum[CatalanNumber[k]^2, {k, 1, n}], 20] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

%t Accumulate[CatalanNumber[Range[1, 20]]^2] (* _Vincenzo Librandi_, Jul 01 2016 *)

%o (Magma) [&+[Catalan(i)^2: i in [1..n]]: n in [1..20]]; // _Vincenzo Librandi_, Jul 01 2016

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

%K nonn

%O 1,2

%A _Alexander Adamchuk_, Nov 15 2009

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