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A160466 Row sums of the Eta triangle A160464.

%I A160466 #10 Jun 02 2025 01:40:09
%S A160466 -1,-9,-87,-2925,-75870,-2811375,-141027075,-18407924325,
%T A160466 -1516052821500,-153801543183750,-18845978136851250,
%U A160466 -2744283682352086875,-468435979952504313750,-92643070481933918821875
%N A160466 Row sums of the Eta triangle A160464.
%C A160466 It is conjectured that the row sums of the Eta triangle depend on five different sequences.
%C A160466 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 A160466 Rowsums(n) = (-1) * A119951(n-1) * FF(n) for n >= 2.
%F A160466 FF(n) = SF(n) * FF(n-1) for n >= 3 with FF(2) =1.
%F A160466 SF(2*n) = A045896(n-2) / A160467(n) for n >= 2.
%F A160466 SF(2*n+1) = A000466(n) / A043529(n-1) for n >= 1.
%p A160466 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 A160466 # End first program.
%p A160466 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 A160466 # End second program.
%Y A160466 A160464 is the Eta triangle.
%Y A160466 Row sum factors A119951, A000466, A043529, A045896 and A160467.
%K A160466 easy,sign
%O A160466 2,2
%A A160466 _Johannes W. Meijer_, May 24 2009