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 A160466 Row sums of the Eta triangle A160464 6
 -1, -9, -87, -2925, -75870, -2811375, -141027075, -18407924325, -1516052821500, -153801543183750, -18845978136851250, -2744283682352086875, -468435979952504313750, -92643070481933918821875 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS It is conjectured that the row sums of the Eta triangle depend on five different sequences. 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. LINKS FORMULA Rowsums(n) = (-1) * A119951(n-1) * FF(n) for n >= 2. FF(n) = SF(n) * FF(n-1) for n >= 3 with FF(2) =1. SF(2*n) = A045896(n-2) / A160467(n) for n >= 2. SF(2*n+1) = A000466(n) / A043529(n-1) for n >= 1. MAPLE 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); # End first program. 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); # End second program. CROSSREFS A160464 is the Eta triangle. Row sum factors A119951, A000466, A043529, A045896 and A160467. Sequence in context: A152264 A035101 A245491 * A015583 A152266 A260041 Adjacent sequences:  A160463 A160464 A160465 * A160467 A160468 A160469 KEYWORD easy,sign AUTHOR Johannes W. Meijer, May 24 2009 STATUS approved

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Last modified July 26 01:17 EDT 2021. Contains 346294 sequences. (Running on oeis4.)