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A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

%I A361355 #14 Oct 15 2024 16:13:59
%S A361355 1,0,0,0,1,0,0,0,1,0,0,0,15,1,0,0,0,0,75,1,0,0,0,0,735,280,1,0,0,0,0,
%T A361355 0,9345,938,1,0,0,0,0,0,76545,77805,2989,1,0,0,0,0,0,0,1865745,536725,
%U A361355 9285,1,0,0,0,0,0,0,13835745,27754650,3334870,28446,1,0
%N A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.
%H A361355 Andrew Howroyd, <a href="/A361355/b361355.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A361355 Luis Ferroni and Matt Larson, <a href="https://arxiv.org/abs/2303.02253">Kazhdan-Lusztig polynomials of braid matroids</a>, arXiv:2303.02253 [math.CO], 2023.
%H A361355 Nicholas Proudfoot, Yuan Xu, and Ben Young, <a href="https://arxiv.org/abs/2406.04502">On the enumeration of series-parallel matroids</a>, arXiv:2406.04502 [math.CO], 2024.
%F A361355 E.g.f.: A(x,y) = log(1 + B(x,y)) where B(x,y) is the e.g.f. of A361353.
%F A361355 E.g.f.: A(x,y) = log(B(log(1 + x), y)/(1 + x)) where B(x,y) is the e.g.f. of A359985.
%F A361355 T(2*n+1, n+1) = A034941(n).
%F A361355 T(2*n, n+1) = A361282(n).
%e A361355 Triangle begins:
%e A361355   1;
%e A361355   0, 0;
%e A361355   0, 1,  0;
%e A361355   0, 0,  1,   0;
%e A361355   0, 0, 15,   1,     0;
%e A361355   0, 0,  0,  75,     1,     0;
%e A361355   0, 0,  0, 735,   280,     1,    0;
%e A361355   0, 0,  0,   0,  9345,   938,    1, 0;
%e A361355   0, 0,  0,   0, 76545, 77805, 2989, 1, 0;
%e A361355   ...
%o A361355 (PARI) \\ B gives A359985 as e.g.f.
%o A361355 B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
%o A361355 T(n) = {my(v=Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))); vector(#v, n, Vecrev(v[n]/y, n))}
%o A361355 { my(A=T(9)); for(i=1, #A, print(A[i])) }
%Y A361355 Row sums are A007834.
%Y A361355 Cf. A140945, A359985, A361353.
%Y A361355 Cf. A034941, A361282.
%K A361355 nonn,tabl
%O A361355 1,13
%A A361355 _Andrew Howroyd_, Mar 09 2023