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A071300
Numerator of b(n) where b(n+1) = Sum_{k=0..n} b'((n^2-k^2)/n), b(0) = b(1) = 1, and b'(x) = b(x) if x is an integer and is linearly interpolated otherwise.
2
1, 1, 2, 9, 65, 82, 1111, 12707, 44127, 270757, 3324143, 773311279, 583426241, 51327727127, 1207458414241, 251022006941731, 1784247347470303, 542442461832071, 59337844204584969481, 172785053530529793211
OFFSET
0,3
LINKS
EXAMPLE
b(4) = b'(9/3) + b'(8/3) + b'(5/3) + b'(0/3) = b(3) + ((1/3)*b(2) + (2/3)*b(3)) + ((1/3)*b(1) + (2/3)*b(2)) + b(0) = 65/6.
MAPLE
bp:= proc(x) local t; option remember; if x::integer then b(x)
else t:= frac(x); t*b(ceil(x))+(1-t)*b(floor(x))
fi
end proc:
b:= proc(m) local k; option remember; add(bp(((m-1)^2-k^2)/(m-1)), k=0..m-1); end proc:
b(0):= 1: b(1):= 1:
map(numer@b, [$0..20]); # Robert Israel, May 20 2019
MATHEMATICA
bp[x_] := bp[x] = Module[{t}, If[IntegerQ[x], b[x],
t = FractionalPart[x]; t*b[Ceiling[x]] + (1-t)*b[Floor[x]]]];
b[m_] := b[m] = Sum[bp[((m-1)^2 - k^2)/(m-1)], {k, 0, m-1}];
b[0] = 1; b[1] = 1;
Map[Numerator@b@#&, Range[0, 20]] (* Jean-François Alcover, Jan 12 2024, after Robert Israel *)
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 10 2002
EXTENSIONS
More terms from David Wasserman
Edited by N. J. A. Sloane, May 20 2019
STATUS
approved