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A157100
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Transform of Catalan numbers whose Hankel transform satisfies the Somos-4 recurrence.
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2
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1, 2, 3, 6, 14, 37, 105, 312, 956, 2996, 9554, 30897, 101083, 333947, 1112497, 3732956, 12605030, 42800318, 146046820, 500555448, 1722402304, 5948047170, 20607691518, 71610355541, 249520257107, 871614139397, 3051737703527
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1+x)*c(x*(1-x^2)), c(x) the g.f. of A000108;
a(n) = Sum_{k=0..n} (-1)^binomial(n-k,2)*binomial(k,floor((n-k)/2))*A000108(k).
Conjecture: (n+1)*a(n) +(-5*n+1)*a(n-1) +2*(2*n-1)*a(n-2) +2*(2*n-7)*a(n-3) +2*(-2*n+7)*a(n-4) = 0. - R. J. Mathar, Feb 05 2015
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MATHEMATICA
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a[n_]:= Sum[(-1)^Binomial[k, 2]*Binomial[n-k, Floor[k/2]]*CatalanNumber[n-k], {k, 0, n}];
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PROG
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(Sage)
def A157100(n): return sum((-1)^binomial(k, 2)*binomial(n-k, k//2)*catalan_number(n-k) for k in (0..n))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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