2,2

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.

Rowsums(n) = (-1)*A120778 (n-2)*A000165 (n-2)*A049606 (n-1) for n = 2, 3, .. .

restart; 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: a:=n-> s1(n): seq(a(n), n=2..nmax);

restart; nmax:=14; for n from 0 to nmax-2 do A120778(n):=numer(sum(binomial(2*k, k)/(k+1) /4^k, k=0..n)) end do: for n from 0 to nmax-2 do A000165(n):=2^n*n! end do: for n from 0 to nmax-2 do A049606(n+1):= denom(2^(n+1)/(n+1)!) end do: for n from 2 to nmax do s2(n):=(-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) end do: a:=n-> s2(n): seq(a(n), n=2..nmax);

A160480 is the Beta triangle.

Row sum factors A120778, A000165 and A049606.

Sequence in context: A024294 A084999 A054593 this_sequence A060608 A003388 A055408

Adjacent sequences: A160478 A160479 A160480 this_sequence A160482 A160483 A160484

easy,sign

Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009