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A343583 a(n) = (1/2)*Li_{-n-1}(1/2) - Li_{-n}(1/2), where Li_{n}(x) is the polylogarithm function. 1
0, 1, 7, 49, 391, 3601, 37927, 451249, 5995591, 88073041, 1418137447, 24846302449, 470675213191, 9587626273681, 209000505036967, 4855088300025649, 119739457665173191, 3124793129198573521, 86030517992814720487, 2492084621605727380849, 75769449406015305475591 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(n) = n! * [x^n] (exp(2*x) - exp(x))/(exp(x) - 2)^2.

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.

a(n) = 2^n*(Epoly(n+1, 1/2) - Epoly(n, 1/2)) where Epoly(n, x) are the Eulerian1 polynomials.

a(n) = A000629(n+1)/2 - A000629(n).

a(n) + 1 = A274273(n).

MAPLE

E1poly := (n, x) -> add(combinat:-eulerian1(n, k)*x^k, k=0..n):

seq(2^n*(E1poly(n+1, 1/2) - E1poly(n, 1/2)), n=0..20);

MATHEMATICA

a[n_] := (PolyLog[-1 - n, 1/2] - 2 PolyLog[-n, 1/2])/2;

Table[a[n], {n, 0, 20}]

PROG

(PARI) a(n) = if (n, polylog(-n-1, 1/2)/2 - polylog(-n, 1/2), 0); \\ Michel Marcus, Apr 26 2021

CROSSREFS

Cf. A000629, A028246, A274273.

Sequence in context: A324353 A349781 A199554 * A221462 A090016 A005924

Adjacent sequences: A343580 A343581 A343582 * A343584 A343585 A343586

KEYWORD

nonn

AUTHOR

Peter Luschny, Apr 26 2021

STATUS

approved

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Last modified January 28 18:12 EST 2023. Contains 359895 sequences. (Running on oeis4.)