A096542 Triangle, read by rows, where e.g.f. A(x,y) satisfies: A(x,y) = exp(x*y*A(x,y+1)) and A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)/n!*x^n*y^k.

1, 0, 1, 0, 2, 3, 0, 15, 30, 16, 0, 244, 564, 444, 125, 0, 6885, 17540, 16680, 7320, 1296, 0, 298326, 817470, 877740, 478380, 136590, 16807, 0, 18377191, 53352138, 62582100, 39142600, 14146440, 2873136, 262144, 0, 1525885992, 4645224472

Offset: 0

Author

Paul D. Hanna, Jun 25 2004

Keywords

• nonn
• tabl

Comments

Row sums form A096537.

Main diagonal forms A000272 (labeled trees on n nodes).

Secondary diagonal forms 2*A057500 (labeled connected graphs with n edges and n nodes).

Other diagonals include 3*A096543 and 4*A096544.

Examples

			A(x,y) = exp(x*y*exp(x*(y+1)*exp(x*(y+2)*exp(...exp(x*(n+y)*exp(...))...)))).
Triangle begins:
1;
0, 1;
0, 2, 3;
0, 15, 30, 16;
0, 244, 564, 444, 125;
0, 6885, 17540, 16680, 7320, 1296;
0, 298326, 817470, 877740, 478380, 136590, 16807;
0, 18377191, 53352138, 62582100, 39142600, 14146440, 2873136, 262144;
0, 1525885992, 4645224472, 5837707848, 4032207480, 1692155640, 441093240, 67558680, 4782969; ...
		

Links

• Paul D. Hanna, Rows n = 0..32, flattened.

Crossrefs

Cf. A096537, A000272, A057500, A096543, A096544.

Programs

PARI

{T(n,k)=local(A=exp(x));for(i=1,n,A=exp(x*(n-i+y)*A+x*O(x^n)+y*O(y^k))); n!*polcoeff(polcoeff(A,k,y),n,x)}

Formula

E.g.f. satisfies: A(x, y+1) = log(A(x, y))/(x*y).

T(n, 1) = n*A096537(n).

T(n, n) = (n+1)^(n-1) = A000272(n+1).

T(n, n-1) = 2*A057500(n).