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%I #16 Apr 27 2020 09:28:54
%S 1,3,12,44,166,626,2377,9063,34695,133265,513381,1982763,7674937,
%T 29767223,115655452,450067268,1753894162,6843602438,26734398172,
%U 104548010228,409243597192,1603372802888,6286998311062,24670701224714,96877958811586,380673221064366
%N Alternating sum of (n+1)*A000108(n+1).
%C Hankel transform is A331474.
%C Alternating sum of A001791(n+1).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*(k+1)*A000108(k+1).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k+2,k).
%F G.f.: (1 - 2*x - sqrt(1-4*x))/(2*x^2*(1+x)*sqrt(1-4*x)).
%F 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
%F 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
%p 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:
%p seq(simplify(a(n)), n=0..25); # _Peter Luschny_, Jan 18 2020
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k+2,k)); \\ _Michel Marcus_, Jan 18 2020
%Y Cf. A000108, A001791, A054109, A331474.
%K nonn
%O 0,2
%A _Paul Barry_, Jan 17 2020
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