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 A126983 Expansion of 1/(1+x*c(x)), c(x) the g.f. of Catalan numbers A000108. 9
 1, -1, 0, -1, -2, -6, -18, -57, -186, -622, -2120, -7338, -25724, -91144, -325878, -1174281, -4260282, -15548694, -57048048, -210295326, -778483932, -2892818244, -10786724388, -40347919626, -151355847012, -569274150156 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Hankel transform is (-1)^n. Catalan transform of A033999. - R. J. Mathar, Nov 11 2008 LINKS Fung Lam, Table of n, a(n) for n = 0..1500 Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021. FORMULA a(n) = (-1)^n*A064310(n). a(n) = Sum_{k=0..n} A039599(n,k)*(-2)^k. From Philippe Deléham, Nov 15 2009: (Start) a(n) = Sum_{k=0..n} A106566(n,k)*(-1)^k, a(0)=1. a(n) = -A000957(n) for n>0. (End) Recurrence: 2*(n+2)*a(n+2) = (7*n+2)*a(n+1) + 2*(2*n+1)*a(n). - Fung Lam, May 07 2014 a(n) ~ -2^(2n)/sqrt(Pi*n^3)/9. - Fung Lam, May 07 2014 MATHEMATICA Table[(-1/2)^n*(1 + Sum[ CatalanNumber[k]*(-2)^k, {k, 0, n-1}]), {n, 0, 30}] (* G. C. Greubel, Feb 27 2019 *) PROG (PARI) {a(n) = (-1/2)^n*(1+sum(k=0, n-1, (-2)^k*binomial(2*k, k)/(k+1)))}; vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 27 2019 (Magma) [1] cat [(-1/2)^n*(1 +(&+[(-2)^k*Binomial(2*k, k)/(k+1): k in [0..n-1]])): n in [1..30]]; // G. C. Greubel, Feb 27 2019 (Sage) [1] + [(-1/2)^n*(1 +sum((-2)^k*catalan_number(k) for k in (0..n-1))) for n in (1..30)] # G. C. Greubel, Feb 27 2019 (Python) from itertools import count, islice def A126983_gen(): # generator of terms yield from (1, -1, 0) a, c = 0, 1 for n in count(1): yield (a:=-a-(c:=c*((n<<2)+2)//(n+2))>>1) A126983_list = list(islice(A126983_gen(), 20)) # Chai Wah Wu, Apr 27 2023 CROSSREFS Cf. A000108, A000957, A039599, A064310, A106566. Sequence in context: A352076 A209797 A064310 * A104629 A000957 A307496 Adjacent sequences: A126980 A126981 A126982 * A126984 A126985 A126986 KEYWORD sign AUTHOR Philippe Deléham, Mar 21 2007 STATUS approved

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Last modified December 9 15:36 EST 2023. Contains 367693 sequences. (Running on oeis4.)