cp's OEIS Frontend

A329052 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.

%I A329052 #8 Jan 09 2020 19:27:57
%S A329052 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 A329052 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 A329052 45,138,278,356,278,138,45,10,1,1,11,55,192,459,724,724,459,192,55,11,1
%N A329052 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 A329052 The two color classes are not interchangeable. Adjacent nodes cannot have the same color.
%H A329052 Andrew Howroyd, <a href="/A329052/b329052.txt">Table of n, a(n) for n = 0..1325</a>
%e A329052 Array begins:
%e A329052 =======================================================
%e A329052 n\m | 0  1   2    3    4     5     6      7      8
%e A329052 ----+--------------------------------------------------
%e A329052   0 | 1, 1,  1,   1,   1,    1,    1,     1,     1, ...
%e A329052   1 | 1, 2,  3,   4,   5,    6,    7,     8,     9, ...
%e A329052   2 | 1, 3,  6,  10,  15,   21,   28,    36,    45, ...
%e A329052   3 | 1, 4, 10,  21,  38,   62,   95,   138,   192, ...
%e A329052   4 | 1, 5, 15,  38,  82,  158,  278,   459,   716, ...
%e A329052   5 | 1, 6, 21,  62, 158,  356,  724,  1359,  2388, ...
%e A329052   6 | 1, 7, 28,  95, 278,  724, 1690,  3612,  7143, ...
%e A329052   7 | 1, 8, 36, 138, 459, 1359, 3612,  8731, 19404, ...
%e A329052   8 | 1, 9, 45, 192, 716, 2388, 7143, 19404, 48213, ...
%e A329052   ...
%o A329052 (PARI)
%o A329052 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 A329052 R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
%o A329052 P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(EulerXY(s))}
%o A329052 { my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }
%Y A329052 Main diagonal is A329055.
%Y A329052 Antidiagonal sums are A329053.
%Y A329052 The equivalent array for labeled nodes is A328887.
%Y A329052 Cf. A329054.
%K A329052 nonn,tabl
%O A329052 0,5
%A A329052 _Andrew Howroyd_, Nov 02 2019