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A360057
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a(n) = Sum_{k=0..n} binomial(n+4*k+4,n-k) * Catalan(k).
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4
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1, 6, 27, 125, 644, 3643, 21974, 138395, 898695, 5970927, 40386209, 277127148, 1924349756, 13496536510, 95467320600, 680260392219, 4878382821267, 35182209381590, 255000022472565, 1856501085686340, 13570366067586294, 99554601986349471, 732756800760507312
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = binomial(n+4,4) + Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^5 + x * A(x)^2.
G.f.: 2 / ( (1-x)^5 * (1 + sqrt( 1 - 4*x/(1-x)^5 )) ).
D-finite with recurrence (n+1)*a(n) +2*(-5*n+1)*a(n-1) +(19*n-11)*a(n-2) +20*(-n+2)*a(n-3) +15*(n-3)*a(n-4) +6*(-n+4)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
(PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1-4*x/(1-x)^5))))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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