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A167892 a(n) = Sum_{k=1..n} Catalan(k)^2. 3
1, 5, 30, 226, 1990, 19414, 203455, 2248355, 25887399, 307993015, 3763786811, 47032778955, 598933188955, 7751562502555, 101741582076580, 1351906409905480, 18159677984049580, 246298405721739580, 3369517588450715680, 46457194476711692080 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

CatalanNumber[k] = (2k)!/k!/(k+1)! = Binomial[2k,k]/(k+1).

REFERENCES

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Catalan Number

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

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A137965 A129695 A110521 * A144498 A201368 A072213

Adjacent sequences:  A167889 A167890 A167891 * A167893 A167894 A167895

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Nov 15 2009

EXTENSIONS

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

STATUS

approved

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Last modified December 11 12:05 EST 2017. Contains 295876 sequences.