A285935 Square array a(n, m) read by antidiagonals whose g.f. is 1 / ((1-x)^2 * (1-y)^2 - x*y).

1, 2, 2, 3, 5, 3, 4, 10, 10, 4, 5, 18, 26, 18, 5, 6, 30, 58, 58, 30, 6, 7, 47, 116, 153, 116, 47, 7, 8, 70, 214, 354, 354, 214, 70, 8, 9, 100, 371, 746, 931, 746, 371, 100, 9, 10, 138, 612, 1464, 2204, 2204, 1464, 612, 138, 10, 11, 185, 969, 2714, 4816, 5794

Offset: 0

Author

Michael Somos, Jun 14 2017

Keywords

• nonn
• tabl

Examples

			a(n,m) 0   1   2   3
----+--- --- --- ---
0   |  1   2   3   4
1   |  2   5  10  18
2   |  3  10  26  58
3   |  4  18  58 153
		

Crossrefs

Cf. A113067, A177787.

Programs

Mathematica

a[n_, m_] := SeriesCoefficient[1/((1-x)^2*(1-y)^2-x*y), {x, 0, n}, {y, 0, m}];
Table[a[n-m, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 15 2017 *)

PARI

{a(n, m) = if( n<0 || m<0, 0, polcoeff( polcoeff( -1/(x*y-sqr(1-x-y+x*y))*(1+x*O(x^n))*(1+y*O(y^k)), n), m))};

Formula

G.f. Sum_{n>=0, m>=0} a(n, m) * x^n * y^m = 1 / ((1-x)^2 * (1-y)^2 - x*y).

T(n, k) := a(n-k, k) where 0 <= k <= n.

a(n, m) = a(m, n) = T(n+m, n), T(n, 0) = a(n, 0) = n+1, if n>=0, m>=0.

Row sums are (-1)^(n+1) * A113067(n+1).

T(n, 1) = A177787(n+1).