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A331473
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Alternating sum of (n+1)*A000108(n+1).
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1
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1, 3, 12, 44, 166, 626, 2377, 9063, 34695, 133265, 513381, 1982763, 7674937, 29767223, 115655452, 450067268, 1753894162, 6843602438, 26734398172, 104548010228, 409243597192, 1603372802888, 6286998311062, 24670701224714, 96877958811586, 380673221064366
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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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a(n) = Sum_{k=0..n} (-1)^(n-k)*(k+1)*A000108(k+1).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k+2,k).
G.f.: (1 - 2*x - sqrt(1-4*x))/(2*x^2*(1+x)*sqrt(1-4*x)).
a(n) = binomial(2*n+4, n+1)*hypergeom ([1, n+5/2, n+3], [n+2, n+4], -4) + (-1)^n*(3*sqrt(5) - 5)/10. - Peter Luschny, Jan 18 2020
D-finite with recurrence +(n+2)*a(n) +(-5*n-4)*a(n-1) +2*(n-5)*a(n-2) +4*(2*n-1)*a(n-3)=0. - R. J. Mathar, Apr 27 2020
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MAPLE
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a := n -> binomial(2*n+4, n+1)*hypergeom([1, n+5/2, n+3], [n+2, n+4], -4) + (-1)^n*(3*sqrt(5) - 5)/10:
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k+2, k)); \\ Michel Marcus, Jan 18 2020
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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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