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Triangle of numbers of irreducible Euler sums.
2

%I #21 Aug 24 2023 12:45:48

%S 1,1,1,1,1,1,1,1,1,2,1,2,2,1,1,3,3,1,2,5,3,1,1,5,7,4,1,3,8,9,4,1,1,7,

%T 14,12,5,1,3,14,20,15,5,1,1,9,25,30,18,6,1,4,20,42,40,22,6,1,1,12,42,

%U 66,55,26,7,1

%N Triangle of numbers of irreducible Euler sums.

%H Vincenzo Librandi, <a href="/A011793/b011793.txt">Rows n = 1..100, flattened</a>

%H D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory</a>, arXiv:hep-th/9604128, 1996.

%H R. P. Loh, A. G. Shannon, and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980. [Taking every other row of this triangle gives the triangle (A258708) in Table 3.]

%t t[n_, k_] := (2/(n+k))*Sum[ If[ EvenQ[d], MoebiusMu[d/2]*Binomial[(n+k)/d, (n-k)/d], 0], {d, Intersection[ Divisors[n+k], Divisors[n-k]]}]; t[1, 1] = t[2, 1] = 1;row[1] = row[2] = {1}; row[n_] := Table[t[n, k], {k, 2 - Mod[n, 2], n-1, 2}]; Flatten[ Table[ row[n], {n, 1, 17}]] (* _Jean-François Alcover_, Jun 15 2012, after _David Broadhurst_ *)

%Y Cf. A258708.

%K nonn,tabl,nice

%O 1,10

%A _N. J. A. Sloane_, _David Broadhurst_