cp's OEIS Frontend

A339228 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements using exactly k colors.

%I A339228 #8 Nov 30 2020 08:56:48
%S A339228 1,2,3,5,22,19,15,146,321,195,48,970,4116,5972,2791,167,6601,48245,
%T A339228 125778,135235,51303,602,46012,546570,2281528,4238415,3609966,1152019,
%U A339228 2256,328188,6118320,38437972,109815445,157612413,111006329,30564075
%N A339228 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements using exactly k colors.
%C A339228 A series configuration is a unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is a unit element or a multiset of two or more series configurations. T(n, k) is the number of series or parallel configurations with n unit elements of k colors using each color at least once.
%e A339228 Triangle begins:
%e A339228     1;
%e A339228     2,     3;
%e A339228     5,    22,     19;
%e A339228    15,   146,    321,     195;
%e A339228    48,   970,   4116,    5972,    2791;
%e A339228   167,  6601,  48245,  125778,  135235,   51303;
%e A339228   602, 46012, 546570, 2281528, 4238415, 3609966, 1152019;
%e A339228   ...
%o A339228 (PARI) \\ R(n,k) gives colorings using at most k colors as a vector.
%o A339228 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339228 R(n,k)={my(Z=k*x, p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); Vec(p)}
%o A339228 M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
%o A339228 {my(T=M(8)); for(n=1, #T~, print(T[n, 1..n]))}
%Y A339228 Columns 1..2 are A003430, A339227.
%Y A339228 Row sums are A339229.
%Y A339228 Main diagonal is A048172.
%Y A339228 Cf. A339226, A339228.
%K A339228 nonn,tabl
%O A339228 1,2
%A A339228 _Andrew Howroyd_, Nov 28 2020