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%I #36 Nov 30 2022 12:39:44
%S 1,2,6,31,227,1991,19415,203456,2248356,25887400,307993016,3763786812,
%T 47032778956,598933188956,7751562502556,101741582076581,
%U 1351906409905481,18159677984049581,246298405721739581
%N Partial sums of squares of Catalan numbers (A000108).
%C 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]
%C 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]
%H Michael De Vlieger, <a href="/A094639/b094639.txt">Table of n, a(n) for n = 0..838</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 Joel E. Cohen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Cohen/cohen13.html">Variance Functions of Asymptotically Exponentially Increasing Integer Sequences Go Beyond Taylor's Law</a>, J. Int. Seq., Vol. 25 (2022), Article 22.9.3.
%F a(n) = Sum_{k=0..n} ((2k)!/(k!)^2/(k+1))^2. - _Alexander Adamchuk_, Feb 16 2008
%F 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 ].
%F 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
%F a(n) ~ 2^(4*n+4) /(15*Pi*n^3). - _Vaclav Kotesovec_, Jul 01 2016
%t Accumulate[CatalanNumber[Range[0,20]]^2] (* _Harvey P. Dale_, May 01 2011 *)
%Y Cf. A000108, A094638, A014137, A001246, A033536, A000984, A006134, A082894, A002897, A079727.
%K easy,nonn
%O 0,2
%A _André F. Labossière_, May 27 2004
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