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.

1, 1, 2, 9, 65, 82, 1111, 12707, 44127, 270757, 3324143, 773311279, 583426241, 51327727127, 1207458414241, 251022006941731, 1784247347470303, 542442461832071, 59337844204584969481, 172785053530529793211

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..350

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

Cf. A071298, A071299, A071301.

Sequence in context: A048801 A152915 A168383 * A062395 A352986 A099975

Adjacent sequences: A071297 A071298 A071299 * A071301 A071302 A071303

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