%I #8 Sep 17 2012 12:45:52
%S -1,-9,-87,-2925,-75870,-2811375,-141027075,-18407924325,
%T -1516052821500,-153801543183750,-18845978136851250,
%U -2744283682352086875,-468435979952504313750,-92643070481933918821875
%N Row sums of the Eta triangle A160464
%C It is conjectured that the row sums of the Eta triangle depend on five different sequences.
%C Two Maple algorithms are given. The first one gives the row sums according to the Eta triangle A160464 and the second one gives the row sums according to our conjecture.
%F Rowsums(n) = (-1) * A119951(n-1) * FF(n) for n >= 2.
%F FF(n) = SF(n) * FF(n-1) for n >= 3 with FF(2) =1.
%F SF(2*n) = A045896(n-2) / A160467(n) for n >= 2.
%F SF(2*n+1) = A000466(n) / A043529(n-1) for n >= 1.
%p nmax:=15; c(2) := -1/3: for n from 3 to nmax do c(n):=(2*n-2)*c(n-1)/(2*n-1)-1/ ((n-1)*(2*n-1)) end do: for n from 2 to nmax do GCS(n-1) := ln(1/(2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2); p(n):=2^(-GCS(n-1))*(2*n-1)!; ETA(n, 1) := p(n)*c(n) end do: mmax:=nmax: for m from 2 to mmax do ETA(2, m) := 0 end do: for n from 3 to nmax do for m from 2 to mmax do q(n) := (1+(-1)^(n-3)*(floor(ln(n-1)/ln(2)) - floor(ln(n-2)/ln(2)))): ETA(n, m) := q(n)*(-ETA(n-1, m-1)+(n-1)^2*ETA(n-1, m)) end do end do: for n from 2 to nmax do s1(n):=0: for m from 1 to n-1 do s1(n) := s1(n) + ETA(n, m) end do end do: seq(s1(n), n=2..nmax);
%p # End first program.
%p nmax:=nmax; A160467 := proc(n): denom(4*(4^n-1)*bernoulli(2*n)/n) end: A043529 := proc(n): ceil(frac(log[2](n+1))+1) end proc: A000466 := proc(n): 4*n^2-1 end proc: A045896 := proc(n): denom((n)/((n+1)*(n+2))) end proc: A119951 := proc(n) : numer(sum(((2*k1)!/(k1!*(k1+1)!))/2^(2*(k1-1)), k1=1..n)) end proc: for n from 1 to nmax do SF(2*n+1):= A000466(n)/A043529(n-1); SF(2*n+2) := A045896(n-1)/A160467(n+1) end do: FF(2):=1: for n from 3 to nmax do FF(n) := SF(n) * FF(n-1) end do: for n from 2 to nmax do s2(n):= (-1)*A119951(n-1)*FF(n) end do: seq(s2(n), n=2..nmax);
%p # End second program.
%Y A160464 is the Eta triangle.
%Y Row sum factors A119951, A000466, A043529, A045896 and A160467.
%K easy,sign
%O 2,2
%A _Johannes W. Meijer_, May 24 2009