A394552 a(n) = n! * [x^n] exp(-Li_{-3}(x)), where Li_{-3}(x) = x*(1+4x+x^2)/(1-x)^4 is the polylogarithm of order -3.

1, -1, -15, -115, -215, 13299, 340081, 5017865, 24029265, -1498877785, -69427044719, -1858615904091, -29317242601415, 246659658348635, 42853508822253585, 2099027490739074209, 68641291277634917281, 1272245656208923310415

Offset

0,3

Comments

By the Truncation-Exactness Theorem (Phalak 2026), if P_n(x) =[exp(-Li_{-3}(x))]_{deg<=n} has roots rho_1,...,rho_n, then Sum_{i=1..n} rho_i^{-k} = k^4 exactly for k = 1,...,n.

Links

Yogesh G. Phalak, Table of n, a(n) for n = 0..400

Yogesh Phalak, Truncated Plethystic Exponentials Preserve Power Sum Constraints, arXiv:2603.28828 [math.NT], 2026.

Formula

a(n) = n!*b(n) where b(0) = 1; b(n) = -(1/n) * Sum_{j=1..n} j^4 * b(n-j) for n >= 1.

E.g.f.: exp(-Li_{-3}(x)) = exp(-x*(1+4*x+x^2)/(1-x)^4).

D-finite with recurrence a(n) +(-5*n+6)*a(n-1) +(10*n-9)*(n-1)*a(n-2) -(n-1)*(n-2)*(10*n-41)*a(n-3) +(n-1)*(n-2)*(n-3)*(5*n-19)*a(n-4) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Mar 24 2026

Example

For n=1: b(1) = -(1/1) * 1^4 * b(0) = -1.
For n=2: b(2) = -(1/2) * (1^4 * b(1) + 2^4 * b(0)) = -(1/2)*(-1+16) = -15/2, and 2! * (-15/2) = -15, confirming a(2) is an integer.
For n=5: the roots of P_5(x) = [exp(-Li_{-3}(x))]_{deg<=5} satisfy Sum_{i=1..5} rho_i^{-k} = k^4 exactly for k=1,2,3,4,5.

Prog

(PARI) first(nn)= my(x='x+O('x^nn)); Vec(serlaplace(exp(-x*(1+4*x+x^2)/(1-x)^4))); \\ Ruud H.G. van Tol, Apr 04 2026

Crossrefs

Cf. A293116 (similar with s=0, alpha_k=k), A318215 (similar with s=-1, alpha_k=k^2).

Sequence in context: A044347 A113527 A044728 * A091347 A125352 A126510

Adjacent sequences: A394549 A394550 A394551 * A394553 A394554 A394555

Keyword

sign,easy

Author

Yogesh G. Phalak, Mar 24 2026

Status

approved