A160481 - OEIS

%I #42 Jun 29 2024 13:31:00

%S -1,-10,-264,-13392,-1111680,-137030400,-23500108800,-5351202662400,

%T -1562069156659200,-568747270103040000,-252681700853514240000,

%U -134539938778433126400000,-84573370199475510312960000,-61972704966344777143418880000,-52361960516341326660973363200000

%N Row sums of the Beta triangle A160480.

%C 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 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 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 Rowsums(n) = (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) for n >= 2.

%F 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 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 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 # End first program

%p 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 # End second program

%t 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 Table[Sum[BETA[n, m], {m, 1, n - 1}], {n, 2, 14}] (* _Jean-François Alcover_, Dec 13 2017 *)

%Y A160480 is the Beta triangle.

%Y Row sum factors A120778, A000165 and A049606.

%Y Cf. A002474, A009445, A129890.

%K easy,sign,uned

%O 2,2

%A _Johannes W. Meijer_, May 24 2009, Sep 19 2012

%E a(15)-a(16) from _Stefano Spezia_, Jun 28 2024