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A189733
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Denominator of B(0,n) where B(n,n)=0, B(n-1,n) = (-1)^(n+1)/n, and B(m,n) = B(m+1,n-1) + B(m,n-1), n >= 0, m >= 0, is an array of fractions.
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4
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1, 1, 1, 2, 1, 6, 1, 4, 3, 5, 1, 12, 1, 7, 5, 8, 1, 18, 1, 10, 7, 11, 1, 24, 1, 13, 9, 14, 1, 30, 1, 16, 11, 17, 1, 36, 1, 19, 13, 20, 1, 42, 1, 22, 15, 23, 1, 48, 1, 25, 17, 26, 1, 54, 1, 28, 19, 29, 1, 60, 1, 31, 21, 32, 1, 66, 1, 34, 23, 35, 1, 72, 1, 37, 25, 38, 1, 78, 1, 40, 27
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OFFSET
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0,4
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COMMENTS
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The square array B(m,n) is defined by values on the diagonal and first subdiagonal and the recurrence of building first differences. It begins in row m=0 as:
0, 1/1, 1/1, 1/2, 0, -1/6, ...
1/1, 0, -1/2, -1/2, -1/6, 1/6, ...
-1/1, -1/2, 0, 1/3, 1/3, 1/12, ...
1/2, 1/2, 1/3, 0, -1/4, -1/4, ...
0, -1/6, -1/3, -1/4, 0, 1/5, ...
-1/6, -1/6, 1/12, 1/4, 1/5, 0, ...
The inverse binomial transform of the first row B(0,n) is the first column up to a sign: B(n,0) = (-1)^(n+1)*B(0,n). In this sense, B(0,n) is an eigensequence of the binomial transform.
B(0,n) = 0, 1/1, 1/1, 1/2, 0, -1/6, 0, 1/4, 1/3, 1/5, 0, -1/12, 0, 1/7, 1/5, 1/8, 0, -1/18, 0, 1/10, 1/7, 1/11, ...
It appears that the sequence of numerators in the first row is 6-periodic: 0, 1, 1, 1, 0, -1.
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LINKS
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FORMULA
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a(n) = denominator(B(0,n)). Conjecture: a(6*n)=1, a(1+6*n)=1+3*n, a(2+6*n)=1+2*n, a(3+6*n)=2+3*n, a(4+6*n)=1, a(5+6*n)=6+6*n. a(n) = 2*a(n-6) - a(n-12).
Empirical g.f.: (1 + x + x^2 + 2*x^3 + x^4 + 6*x^5 - x^6 + 2*x^7 + x^8 + x^9 - x^10) / ((1 - x)^2 * (1 + x)^2 * (1 - x + x^2)^2 * (1 + x + x^2)^2). - Colin Barker, Nov 11 2016
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MAPLE
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B := proc(m, n) option remember; if m=n then 0; elif n = m+1 then (-1)^(n+1)/n ; elif n > m then procname(m, n-1)+procname(m+1, n-1) ; elif n < m then procname(m-1, n+1)-procname(m-1, n) ; end if; end proc:
A189733 := proc(n) denom(B(0, n)) ; end proc:
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MATHEMATICA
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b[m_, n_] := b[m, n] = Which[m == n, 0, n == m+1, (-1)^(n+1)/n, n > m, b[m, n-1] + b[m+1, n-1], n < m, b[m-1, n+1] - b[m-1, n]]; a[n_] := b[0, n] // Denominator; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 07 2013 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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