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Array read by antidiagonals: T(n,m) is the number of unlabeled bicolored acyclic graphs with n nodes of one color and m of the other.
3

%I #8 Jan 09 2020 19:27:57

%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,15,21,15,6,1,1,7,21,

%T 38,38,21,7,1,1,8,28,62,82,62,28,8,1,1,9,36,95,158,158,95,36,9,1,1,10,

%U 45,138,278,356,278,138,45,10,1,1,11,55,192,459,724,724,459,192,55,11,1

%N Array read by antidiagonals: T(n,m) is the number of unlabeled bicolored acyclic graphs with n nodes of one color and m of the other.

%C The two color classes are not interchangeable. Adjacent nodes cannot have the same color.

%H Andrew Howroyd, <a href="/A329052/b329052.txt">Table of n, a(n) for n = 0..1325</a>

%e Array begins:

%e =======================================================

%e n\m | 0 1 2 3 4 5 6 7 8

%e ----+--------------------------------------------------

%e 0 | 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

%e 2 | 1, 3, 6, 10, 15, 21, 28, 36, 45, ...

%e 3 | 1, 4, 10, 21, 38, 62, 95, 138, 192, ...

%e 4 | 1, 5, 15, 38, 82, 158, 278, 459, 716, ...

%e 5 | 1, 6, 21, 62, 158, 356, 724, 1359, 2388, ...

%e 6 | 1, 7, 28, 95, 278, 724, 1690, 3612, 7143, ...

%e 7 | 1, 8, 36, 138, 459, 1359, 3612, 8731, 19404, ...

%e 8 | 1, 9, 45, 192, 716, 2388, 7143, 19404, 48213, ...

%e ...

%o (PARI)

%o EulerXY(A)={my(j=serprec(A,x)); exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))}

%o R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};

%o P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(EulerXY(s))}

%o { my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }

%Y Main diagonal is A329055.

%Y Antidiagonal sums are A329053.

%Y The equivalent array for labeled nodes is A328887.

%Y Cf. A329054.

%K nonn,tabl

%O 0,5

%A _Andrew Howroyd_, Nov 02 2019