A096542 - OEIS

%I #11 Dec 19 2017 13:22:00

%S 1,0,1,0,2,3,0,15,30,16,0,244,564,444,125,0,6885,17540,16680,7320,

%T 1296,0,298326,817470,877740,478380,136590,16807,0,18377191,53352138,

%U 62582100,39142600,14146440,2873136,262144,0,1525885992,4645224472

%N 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.

%C Row sums form A096537.

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

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

%C Other diagonals include 3*A096543 and 4*A096544.

%H Paul D. Hanna, <a href="/A096542/b096542.txt">Rows n = 0..32, flattened.</a>

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

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

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

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

%e A(x,y) = exp(x*y*exp(x*(y+1)*exp(x*(y+2)*exp(...exp(x*(n+y)*exp(...))...)))).

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 2, 3;

%e 0, 15, 30, 16;

%e 0, 244, 564, 444, 125;

%e 0, 6885, 17540, 16680, 7320, 1296;

%e 0, 298326, 817470, 877740, 478380, 136590, 16807;

%e 0, 18377191, 53352138, 62582100, 39142600, 14146440, 2873136, 262144;

%e 0, 1525885992, 4645224472, 5837707848, 4032207480, 1692155640, 441093240, 67558680, 4782969; ...

%o (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)}

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jun 25 2004