%I #17 Apr 26 2021 10:54:15
%S 0,1,7,49,391,3601,37927,451249,5995591,88073041,1418137447,
%T 24846302449,470675213191,9587626273681,209000505036967,
%U 4855088300025649,119739457665173191,3124793129198573521,86030517992814720487,2492084621605727380849,75769449406015305475591
%N a(n) = (1/2)*Li_{-n-1}(1/2) - Li_{-n}(1/2), where Li_{n}(x) is the polylogarithm function.
%F a(n) = n! * [x^n] (exp(2*x) - exp(x))/(exp(x) - 2)^2.
%F a(n) = Sum_{k=0..n} T(n, k)*k, where T(n, k) = Sum_{j=0..n} Eulerian1(n, j)* binomial(n-j, n-k) is the (0, 0)-based variant of A028246.
%F a(n) = 2^n*(Epoly(n+1, 1/2) - Epoly(n, 1/2)) where Epoly(n, x) are the Eulerian1 polynomials.
%F a(n) = A000629(n+1)/2 - A000629(n).
%F a(n) + 1 = A274273(n).
%p E1poly := (n, x) -> add(combinat:-eulerian1(n,k)*x^k, k=0..n):
%p seq(2^n*(E1poly(n+1, 1/2) - E1poly(n, 1/2)), n=0..20);
%t a[n_] := (PolyLog[-1 - n, 1/2] - 2 PolyLog[-n, 1/2])/2;
%t Table[a[n], {n, 0, 20}]
%o (PARI) a(n) = if (n, polylog(-n-1, 1/2)/2 - polylog(-n, 1/2), 0); \\ _Michel Marcus_, Apr 26 2021
%Y Cf. A000629, A028246, A274273.
%K nonn
%O 0,3
%A _Peter Luschny_, Apr 26 2021
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