2,2

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.

FF(n) = (-1)*Rowsums(n) / A119951(n-1) for n = 2, 3, 4, .. .

SF(n) = FF(n)/FF(n-1) for n = 3, 4, 5, .. .

SF(2*n) = A045896(n-2) / A160467(n) for n = 2, 3, 4, .. .

SF(2*n+1) = A000466(n) / A043529(n-1) for n = 1, 2, 3, .. .

restart; nmax:=16; mmax:=nmax: 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) od: for n from 2 to nmax do p(n):=2^(-GCS(n-1))*(2*n-1)! od: for n from 2 to nmax do ETA(n, 1):=p(n)*c(n) end do: 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: a:=n-> s1(n): seq(a(n), n=2..nmax);

restart; nmax:=16; p:= floor(ln(nmax)/ln(2)): for n from 1 to nmax do A160467(n):=1 end do: for q from 1 to p do for n from 1 to nmax do if n mod 2^q = 0 then A160467(n):=2^(q-1) end if: end do: end do: for n from 1 to nmax do A160467(n):=A160467(n) end do: A043529(0):=1: for n from 1 to nmax do A043529(n):= denom((abs(floor(ln(n+1)/ln(2))+((n)mod 2)-floor(ln(n)/ln(2))-((n-1) mod 2)))/2) end do: for n from 1 to nmax do A000466(n):=4*n^2-1 end do: for n from 0 to nmax do A045896 (n):=denom((n)/((n+1)*(n+2))) end do: for n from 1 to nmax do SF(2*n+1):= A000466(n)/A043529(n-1) end do: for n from 2 to nmax do SF(2*n):=A045896 (n-2)/A160467(n) end do: for n from 1 to nmax-1 do FF(n) end do: for n from 1 to nmax do A119951(n):= numer(sum(((2*k)!/(k!*(k+1)!))/2^(2*(k-1)), k=1..n)) end do: FF(2):=1: for n from 3 to nmax do FF(n):=FF(n-1)*SF(n) end do: for n from 2 to nmax do s2(n):= -A119951(n-1)*FF(n) end do: a:=n-> s2(n): seq(a(n), n=2..nmax);

A160464 is the Eta triangle.

Row sum factors A119951, A000466, A043529, A045896, A160467

Sequence in context: A153191 A152264 A035101 this_sequence A015583 A152266 A084022

Adjacent sequences: A160463 A160464 A160465 this_sequence A160467 A160468 A160469

easy,sign

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