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)).

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, 192, 20, 1, 21, 168, 644, 1225, 1085, 371, 31, 1, 24, 228, 1096, 2834, 3880, 2588, 696, 49, 1, 27, 297, 1719, 5652, 10656, 11097, 5823, 1278, 78, 1, 30, 375, 2540, 10165

Offset: 0

Author

Vladeta Jovovic, Jun 18 2001

Keywords

• easy
• nonn
• tabl

Comments

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

See Riordan 1954 page 21 equation (24). - Michael Somos, Aug 26 2015

Examples

			Triangle begins:
1,
1,3,
1,6,5,
1,9,18,6,
1,12,42,44,9,
1,15,75,145,95,13,
1,18,117,336,420,192,20,
1,21,168,644,1225,1085,371,31,
1,24,228,1096,2834,3880,2588,696,49,
1,27,297,1719,5652,10656,11097,5823,1278,78,
1,30,375,2540,10165,24626,35045,29380,12535,2310,125,
... (from _N. J. A. Sloane_, Jun 28 2015)
Sum_{n, k} T(n, k) u^n t^k = 1 + (1 + 3*t)*u + (1 + 6*t + 5*t^2)*u^2 + ...
		

References

• R. P. Stanley, Enumerative Combinatorics I, Example 4.7.17.

Links

• J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] See Table 1.

Crossrefs

Cf. A000183, row sums: A061703, third column: A000338, fourth column: A000561, fifth column: A000562, sixth column: A000563, seventh column: A000564, eighth column: A000565.

Programs

Mathematica

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 *)

Extensions

Edited by N. J. A. Sloane, Jun 28 2015