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A361353 Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.

%I A361353 #16 Oct 15 2024 17:11:43
%S A361353 1,0,1,0,1,1,0,0,5,1,0,0,15,16,1,0,0,0,175,42,1,0,0,0,735,1225,99,1,0,
%T A361353 0,0,0,16065,6769,219,1,0,0,0,0,76545,204400,32830,466,1,0,0,0,0,0,
%U A361353 2747745,2001230,147466,968,1,0,0,0,0,0,13835745,56143395,16813720,632434,1981,1
%N A361353 Triangle read by rows: T(n,k) is the number of simple quasi series-parallel matroids on [n] with rank k, 1 <= k <= n.
%C A361353 See Table 2 in the Ferroni/Larson reference.
%H A361353 Andrew Howroyd, <a href="/A361353/b361353.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A361353 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 A361353 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 A361353 E.g.f.: A(x,y) = B(log(1 + x), y)/(1 + x) - 1 where B(x,y) is the e.g.f. of A359985.
%e A361353 Triangle begins:
%e A361353   1;
%e A361353   0, 1;
%e A361353   0, 1,  1;
%e A361353   0, 0,  5,   1;
%e A361353   0, 0, 15,  16,     1;
%e A361353   0, 0,  0, 175,    42,      1;
%e A361353   0, 0,  0, 735,  1225,     99,     1;
%e A361353   0, 0,  0,   0, 16065,   6769,   219,   1;
%e A361353   0, 0,  0,   0, 76545, 204400, 32830, 466, 1;
%e A361353   ...
%o A361353 (PARI) \\ B gives A359985 as e.g.f.
%o A361353 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 A361353 T(n) = {[Vecrev(p/y) | p<-Vec(serlaplace(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x) - 1))]}
%o A361353 { my(A=T(9)); for(i=1, #A, print(A[i])) }
%Y A361353 Row sums are A361354.
%Y A361353 Cf. A140945, A359985, A361355.
%K A361353 nonn,tabl
%O A361353 1,9
%A A361353 _Andrew Howroyd_, Mar 09 2023