This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381681 #38 Apr 29 2025 23:31:44
%S A381681 0,1,2,7,30,159,998,7251,59862,553591,5669406,63698427,779065694,
%T A381681 10304068863,146547757014,2230287456259,36165665815878,
%U A381681 622513383121671,11336090988469742,217741030441959051,4399571340398826126,93286012779568250767,2071087588405552461414,48048511292938827392403
%N A381681 a(n) is one of two integer components (with A000254) used in computing the inverse second moment of X+n, where X~Poisson(1).
%C A381681 Analog of A093344 (with alternating terms in inner summation).
%H A381681 Michael R. Powers, <a href="https://arxiv.org/abs/2503.02054">Conjunctions of Three "Euler Constants" in Poisson-Related Expressions</a>, arXiv:2503.02054 [math.NT], 2025.
%F A381681 a(n) = n! * Sum_{i=1..n} (1/i)*Sum_{j=0..i-1} (-1)^j/j!.
%e A381681 If X~Poisson(1), then E[(X+n)^(-2)] = (-1)^n * {(n-1)! * [-Ei(1)+gamma] - A000254(n-1) + e*a(n-1)}/e for n = 1,2,... where gamma is Euler's constant.
%o A381681 (PARI) a(n) = n! * sum(i=1, n, (1/i)*sum(j=0, i-1, (-1)^j/j!)); \\ _Michel Marcus_, Mar 07 2025
%Y A381681 A093344 gives one of two integer components (with A000254) used in computing the alternating inverse second moment of X+n for X~Poisson(1).
%K A381681 nonn
%O A381681 0,3
%A A381681 _Michael R. Powers_, Mar 05 2025