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A259342 Irregular triangle read by rows: T(n,k) = number of equivalence classes of binary sequences of length n containing exactly 2k ones.

%I A259342 #21 Jul 01 2017 18:05:52
%S A259342 1,1,1,1,1,1,2,1,1,2,1,1,3,3,1,1,3,4,1,1,4,8,4,1,1,4,10,7,1,1,5,16,16,
%T A259342 5,1,1,5,20,26,10,1,1,6,29,50,29,6,1,1,6,35,76,57,14,1,1,7,47,126,126,
%U A259342 47,7,1,1,7,56,185,232,111,19,1,1,8,72,280,440,280,72,8,1
%N A259342 Irregular triangle read by rows: T(n,k) = number of equivalence classes of binary sequences of length n containing exactly 2k ones.
%H A259342 Alois P. Heinz, <a href="/A259342/b259342.txt">Rows n = 1..200, flattened</a>
%H A259342 W. D. Hoskins and Anne Penfold Street, <a href="http://dx.doi.org/10.1017/S1446788700017547">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
%H A259342 W. D. Hoskins and A. P. Street, <a href="/A005513/a005513_1.pdf">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
%F A259342 Theorem 1 of Hoskins-Street gives an explicit formula.
%e A259342 Triangle begins:
%e A259342   1;
%e A259342   1, 1;
%e A259342   1, 1;
%e A259342   1, 2,  1;
%e A259342   1, 2,  1;
%e A259342   1, 3,  3,   1;
%e A259342   1, 3,  4,   1;
%e A259342   1, 4,  8,   4,   1;
%e A259342   1, 4, 10,   7,   1;
%e A259342   1, 5, 16,  16,   5,  1;
%e A259342   1, 5, 20,  26,  10,  1;
%e A259342   1, 6, 29,  50,  29,  6, 1;
%e A259342   1, 6, 35,  76,  57, 14, 1;
%e A259342   1, 7, 47, 126, 126, 47, 7, 1;
%e A259342   ...
%p A259342 with(numtheory):
%p A259342 T:= (n, k)-> (add(`if`(irem(2*k*d, n)=0, phi(n/d)
%p A259342      *binomial(d, 2*k*d/n), 0), d=divisors(n))
%p A259342      +n*binomial(iquo(n, 2), k))/(2*n):
%p A259342 seq(seq(T(n, k), k=0..n/2), n=1..20);  # _Alois P. Heinz_, Jun 28 2015
%t A259342 T[n_, k_] := (DivisorSum[n, If[Mod[2k*#, n]==0, EulerPhi[n/#]*Binomial[#, 2k*#/n], 0]&] + n*Binomial[Quotient[n, 2], k])/(2n); Table[T[n, k], {n, 1, 20}, { k, 0, n/2}] // Flatten (* _Jean-François Alcover_, Feb 28 2017, after _Alois P. Heinz_ *)
%Y A259342 Row sums are (essentially) A000011.
%K A259342 nonn,tabf
%O A259342 1,7
%A A259342 _N. J. A. Sloane_, Jun 27 2015