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A160481 Row sums of the Beta triangle A160480. 3
-1, -10, -264, -13392, -1111680, -137030400, -23500108800, -5351202662400, -1562069156659200, -568747270103040000, -252681700853514240000, -134539938778433126400000, -84573370199475510312960000 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..14.

CP Herzog, KW Huang, K Jensen, Universal Entanglement and Boundary Geometry in Conformal Field Theory, arXiv preprint arXiv:1510.00021, 2015

FORMULA

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

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

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

MAPLE

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);

# End first program

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);

# End second program

MATHEMATICA

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;

Table[Sum[BETA[n, m], {m, 1, n - 1}], {n, 2, 14}] (* Jean-Fran├žois Alcover, Dec 13 2017 *)

CROSSREFS

A160480 is the Beta triangle.

Row sum factors A120778, A000165 and A049606.

Sequence in context: A217911 A054593 A290041 * A060608 A003388 A055408

Adjacent sequences:  A160478 A160479 A160480 * A160482 A160483 A160484

KEYWORD

easy,sign

AUTHOR

Johannes W. Meijer, May 24 2009, Sep 19 2012

STATUS

approved

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Last modified February 16 20:45 EST 2019. Contains 320189 sequences. (Running on oeis4.)