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 A160481 #42 Jun 29 2024 13:31:00
%S A160481 -1,-10,-264,-13392,-1111680,-137030400,-23500108800,-5351202662400,
%T A160481 -1562069156659200,-568747270103040000,-252681700853514240000,
%U A160481 -134539938778433126400000,-84573370199475510312960000,-61972704966344777143418880000,-52361960516341326660973363200000
%N A160481 Row sums of the Beta triangle A160480.
%C A160481 It is conjectured that the row sums of the Beta triangle depend on three different sequences. Two Maple algorithms are given. The first one gives the row sums according to the Beta triangle A160480 and the second one gives the row sums according to our conjecture.
%H A160481 Christopher P. Herzog, Kuo-Wei Huang, and Kristan Jensen, <a href="http://arxiv.org/abs/1510.00021">Universal Entanglement and Boundary Geometry in Conformal Field Theory</a>, arXiv preprint arXiv:1510.00021 [hep-th], 2015.
%H A160481 Kuo-Wei Huang, <a href="https://arxiv.org/abs/2406.07458">Resummation of Multi-Stress Tensors in Higher Dimensions</a>, arXiv:2406.07458 [hep-th], 2024. See p. 10.
%F A160481 Rowsums(n) = (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) for n >= 2.
%F A160481 Conjecture: a(n) = (2*n-3)! - 2^(2*n-3)*(n-1)!*(n-2)!, for n >= 2 (gives the first 13 terms). - _Christopher P. Herzog_, Nov 25 2014
%F A160481 Meijer's and Herzog's conjectures can also be written as: a(n) = -A129890(n-2)*A000165(n-2) = A009445(n-2) - A002474(n-2). - _Peter Luschny_, Dec 01 2014
%p A160481 nmax := 14; mmax := nmax: for n from 1 to nmax do BETA(n, n) := 0 end do: m := 1: for n from m+1 to nmax do BETA(n,m) := (2*n-3)^2*BETA(n-1, m)-(2*n-4)! od: for m from 2 to mmax do for n from m+1 to nmax do BETA(n, m) := (2*n-3)^2*BETA(n-1, m) - BETA(n-1, m-1) od: od: for n from 2 to nmax do s1(n) := 0: for m from 1 to n-1 do s1(n) := s1(n) + BETA(n, m) od: od: seq(s1(n), n=2..nmax);
%p A160481 # End first program
%p A160481 nmax := nmax; A120778 := proc(n): numer(sum(binomial(2*k1, k1)/(k1+1) / 4^k1, k1=0..n)) end proc: A000165 := proc(n): 2^n*n! end proc: A049606 := proc(n): denom(2^n/n!) end proc: for n from 2 to nmax do s2(n) := (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) end do: seq(s2(n), n=2..nmax);
%p A160481 # End second program
%t A160481 BETA[2, 1] = -1; BETA[n_, 1] := BETA[n, 1] = (2*n - 3)^2*BETA[n - 1, 1] - (2*n - 4)!; BETA[n_ /; n > 2, m_ /; m > 0] /; 1 <= m <= n := BETA[n, m] = (2*n - 3)^2*BETA[n - 1, m] - BETA[n - 1, m - 1]; BETA[_, _] = 0;
%t A160481 Table[Sum[BETA[n, m], {m, 1, n - 1}], {n, 2, 14}] (* _Jean-François Alcover_, Dec 13 2017 *)
%Y A160481 A160480 is the Beta triangle.
%Y A160481 Row sum factors A120778, A000165 and A049606.
%Y A160481 Cf. A002474, A009445, A129890.
%K A160481 easy,sign,uned
%O A160481 2,2
%A A160481 _Johannes W. Meijer_, May 24 2009, Sep 19 2012
%E A160481 a(15)-a(16) from _Stefano Spezia_, Jun 28 2024