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%I #9 Jun 14 2024 03:16:50
%S 1,1,1,1,3,1,1,7,7,1,1,15,35,15,1,1,31,155,155,31,1,1,63,651,1365,651,
%T 63,1,1,127,2667,10941,10941,2667,127,1,1,255,10795,82215,156597,
%U 82215,10795,255,1,1,511,43435,589135,1988007,1988007,589135,43435,511,1
%N Triangle read by rows: T(n,k) is the number of quasi series-parallel matroids on [n] with rank k, 0 <= k <= n.
%C A quasi series-parallel matroid is a collection of series-parallel matroids. See the Ferroni/Larson reference for a precise definition.
%C The first six rows of this triangle are the same as A022166.
%H 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 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.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 7, 1;
%e 1, 15, 35, 15, 1;
%e 1, 31, 155, 155, 31, 1;
%e 1, 63, 651, 1365, 651, 63, 1;
%e 1, 127, 2667, 10941, 10941, 2667, 127, 1;
%e ...
%o (PARI) \\ Proposition 2.3, 2.8 in Ferroni/Larson, compare A140945.
%o T(n) = {[Vecrev(p) | p<-Vec(serlaplace(exp(x*(y+1) + y*intformal( serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))))]}
%o { my(A=T(8)); for(i=1, #A, print(A[i])) }
%Y Row sums are A359986.
%Y Columns k=0..2 are A000012, A000225, A006095.
%Y Cf. A022166, A140945.
%K nonn,tabl
%O 0,5
%A _Andrew Howroyd_, Mar 08 2023
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