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A094639 Partial sums of squares of Catalan numbers (A000108). 7
1, 2, 6, 31, 227, 1991, 19415, 203456, 2248356, 25887400, 307993016, 3763786812, 47032778956, 598933188956, 7751562502556, 101741582076581, 1351906409905481, 18159677984049581, 246298405721739581 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Koshy and Salmassi give an elementary proof that the only prime Catalan numbers are A000108(2) = 2 and A000108(3) = 5. Franklin T. Adams-Watters showed that the only semiprime Catalan number is A000108(4) = 14. The subsequence of primes in the partial sum of squares of Catalan numbers begins: 2, 31, 227, 101741582076581. [Jonathan Vos Post, May 27 2010]

Conjecture: For any positive integer n, the polynomial P_n(x) = sum_{k = 0}^n(C_k)^2*x^k (with C_k = binomial(2k, k)/(k+1)) is irreducible over the field of rational numbers. [Zhi-Wei Sun, Mar 23 2013]

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

Table of n, a(n) for n=0..18.

FORMULA

a(n) = Sum_{k=0..n} ((2k)!/(k!)^2/(k+1))^2. - Alexander Adamchuk, Feb 16 2008

Sum_{i=1..n} [c(i)]^2 = Sum_{i=1..n} [C(2*i-2, i-1)/i]^2 = (1/(n-1)!)^2 * [ n^C(2*n-4, 1) + {2*C(n-1, 2)}*n^(2*n-5) + {C(n-2, 0) + 4*C(n-2, 1) + 13*C(n-2, 2) + 22*C(n-2, 3) + 12*C(n-2, 4)}*n^C(2*n-6, 1) + {12*C(n-3, 1) + 152*C(n-3, 2) + 458*C(n-3, 3) + 640*C(n-3, 4) + 440*C(n-3, 5) + 120*C(n-3, 6)}*n^(2*n-7) + {40*C(n-4, 0) + 313*C(n-4, 1) + 2332*C(n-4, 2) + 9536*C(n-4, 3) + 21409*C(n-4, 4) + 28068*C(n-4, 5) + 21700*C(n-4, 6) + 9240*C(n-4, 7) + 1680*C(n-4, 8) + ... + C(n-3, 0)*((n-1)!)^2 ].

Recurrence: (n+1)^2*a(n) = (17*n^2 - 14*n + 5)*a(n-1) - 4*(2*n - 1)^2*a(n-2). - Vaclav Kotesovec, Jul 01 2016

a(n) ~ 2^(4*n+4) /(15*Pi*n^3). - Vaclav Kotesovec, Jul 01 2016

MATHEMATICA

Accumulate[CatalanNumber[Range[0, 20]]^2] (* Harvey P. Dale, May 01 2011 *)

CROSSREFS

Cf. A000108, A094638, A014137, A001246, A033536, A000984, A006134, A082894, A002897, A079727.

Sequence in context: A054141 A007710 A275558 * A113719 A018225 A217143

Adjacent sequences:  A094636 A094637 A094638 * A094640 A094641 A094642

KEYWORD

easy,nonn

AUTHOR

André F. Labossière, May 27 2004

STATUS

approved

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Last modified December 15 03:08 EST 2017. Contains 296020 sequences.