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A061702 Triangle T(n,k) defined by Sum_{n >= 0,m >= 0} T(n,m)*x^m*y^n = 1 + y*(1 + 3*x - 4*x^2*y - 3*x^2*y^2 - 3*x^3*y^2 + 4*x^4*y^3)/((1 - y - 2*x*y - x*y^2 + x^3*y^3)*(1 - x*y)).

%I A061702 #17 Aug 26 2015 12:07:49
%S A061702 1,1,3,1,6,5,1,9,18,6,1,12,42,44,9,1,15,75,145,95,13,1,18,117,336,420,
%T A061702 192,20,1,21,168,644,1225,1085,371,31,1,24,228,1096,2834,3880,2588,
%U A061702 696,49,1,27,297,1719,5652,10656,11097,5823,1278,78,1,30,375,2540,10165
%N A061702 Triangle T(n,k) defined by Sum_{n >= 0,m >= 0} T(n,m)*x^m*y^n = 1 + y*(1 + 3*x - 4*x^2*y - 3*x^2*y^2 - 3*x^3*y^2 + 4*x^4*y^3)/((1 - y - 2*x*y - x*y^2 + x^3*y^3)*(1 - x*y)).
%C A061702 It is uncertain if the initial term should be 0 or 1. Both make sense. I have changed the data line to start with 1, in agreement with Riordan (1954). - _N. J. A. Sloane_, Jun 28 2015
%C A061702 See Riordan 1954 page 21 equation (24). - _Michael Somos_, Aug 26 2015
%D A061702 R. P. Stanley, Enumerative Combinatorics I, Example 4.7.17.
%H A061702 J. Riordan, <a href="/A000211/a000211.pdf">Discordant permutations</a>, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] See Table 1.
%e A061702 Triangle begins:
%e A061702 1,
%e A061702 1,3,
%e A061702 1,6,5,
%e A061702 1,9,18,6,
%e A061702 1,12,42,44,9,
%e A061702 1,15,75,145,95,13,
%e A061702 1,18,117,336,420,192,20,
%e A061702 1,21,168,644,1225,1085,371,31,
%e A061702 1,24,228,1096,2834,3880,2588,696,49,
%e A061702 1,27,297,1719,5652,10656,11097,5823,1278,78,
%e A061702 1,30,375,2540,10165,24626,35045,29380,12535,2310,125,
%e A061702 ... (from _N. J. A. Sloane_, Jun 28 2015)
%e A061702 Sum_{n, k} T(n, k) u^n t^k = 1 + (1 + 3*t)*u + (1 + 6*t + 5*t^2)*u^2 + ...
%t A061702 max = 11; f[x_, y_] := 1 + y*(1 + 3*x - 4*x^2*y - 3*x^2*y^2 - 3*x^3*y^2 + 4*x^4*y^3)/((1 - y - 2*x*y - x*y^2 + x^3*y^3)*(1 - x*y)); se = Series[f[x, y], {x, 0, max}, {y, 0, max}]; coes = CoefficientList[se, {x, y}] ; t[n_, k_] := coes[[k, n]]; Flatten[ Table[t[n, k], {n, 1, max}, {k, 1, n}]](* _Jean-François Alcover_, Oct 24 2011 *)
%Y A061702 Cf. A000183, row sums: A061703, third column: A000338, fourth column: A000561, fifth column: A000562, sixth column: A000563, seventh column: A000564, eighth column: A000565.
%K A061702 easy,nonn,tabl
%O A061702 0,3
%A A061702 _Vladeta Jovovic_, Jun 18 2001
%E A061702 Edited by _N. J. A. Sloane_, Jun 28 2015