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%I #43 Apr 27 2023 11:22:54
%S 1,1,0,1,-2,6,-18,57,-186,622,-2120,7338,-25724,91144,-325878,1174281,
%T -4260282,15548694,-57048048,210295326,-778483932,2892818244,
%U -10786724388,40347919626,-151355847012,569274150156
%N Generalized Catalan numbers C(-1; n).
%C See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
%C Unsigned sequence with a(0) := 0 is A000957 (Fine).
%H G. C. Greubel, <a href="/A064310/b064310.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [_N. J. A. Sloane_, Oct 08 2012]
%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%H P. Pagacz and M. Wojtylak, <a href="http://arxiv.org/abs/1310.2122">On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics</a>, arXiv:1310.2122 [math.PR], 2013.
%F a(n) = Sum_{m=0..n-1} (-1)^m*(n-m)*binomial(n-1+m, m)/n.
%F a(n) = ((1/2)^n)*(1 + Sum_{k=0..n-1} C(k)*(-2)^k ), n >= 1, a(0)= 1, with C(n)=A000108(n) (Catalan).
%F G.f.: (1+x*c(-x)/2)/(1-x/2) = 1/(1-x*c(-x)) with c(x) g.f. of Catalan numbers A000108.
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*A106566(n, k). - _Philippe Deléham_, Sep 18 2005
%F (-1)^n*a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k. - _Philippe Deléham_, Mar 13 2007
%F Conjecture: 2*n*a(n) + (7*n-12)*a(n-1) + 2*(-2*n+3)*a(n-2) = 0. - _R. J. Mathar_, Dec 02 2012
%t a[n_]:= (1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]); Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 17 2013 *)
%o (PARI) {a(n) = (1 + sum(k=0, n-1, (-2)^k*binomial(2*k,k)/(k+1)))/2^n};
%o vector(30, n, n--; a(n)) \\ _G. C. Greubel_, Feb 27 2019
%o (Magma) [1] cat [(1 +(&+[(-2)^k*Binomial(2*k,k)/(k+1): k in [0..n-1]]))/2^n: n in [1..30]]; // _G. C. Greubel_, Feb 27 2019
%o (Sage) [1] + [(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1)))/2^n for n in (1..30)] # _G. C. Greubel_, Feb 27 2019
%o (Python)
%o from itertools import count, islice
%o def A064310_gen(): # generator of terms
%o yield from (1,1,0)
%o a, c = 0, 1
%o for n in count(1):
%o yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)*(1 if n&1 else -1)
%o A064310_list = list(islice(A064310_gen(),20)) # _Chai Wah Wu_, Apr 27 2023
%Y Cf. A000108, A000957, A039599, A106566.
%K sign,easy
%O 0,5
%A _Wolfdieter Lang_, Sep 21 2001
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