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A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.

%I A098233 #33 Apr 18 2025 05:49:48
%S A098233 1,1,1,1,1,1,4,7,3,1,1,13,46,47,25,6,1,1,40,295,587,516,235,65,10,1,1,
%T A098233 121,1846,6715,9690,7053,3006,800,140,15,1,1,364,11347,73003,170051,
%U A098233 189458,119211,46795,12201,2170,266,21,1,1,1093,68986,768747
%N A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.
%C A098233 Also gives number T(n, k) of partitions of the multiset {1, 1, 2, 2, ..., n, n} into k nonempty subsets, for 2 <= k <= 2n. - _Marko Riedel_, Jan 22 2023
%D A098233 G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.
%H A098233 Steve Butler, Fan Chung, Jay Cummings, and R. L. Graham, <a href="http://arxiv.org/abs/1504.01426">Juggling card sequences</a>, arXiv:1504.01426 [math.CO], 2015.
%H A098233 L. Comtet, <a href="/A002718/a002718.pdf">Birecouvrements et birevêtements d'un ensemble fini</a>, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]
%H A098233 G. Paquin, <a href="/A038205/a038205.pdf">Dénombrement de multigraphes enrichis</a>, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]
%e A098233 The first few polynomials are:
%e A098233   1,
%e A098233   x^2,
%e A098233   x^2+x^3+x^4,
%e A098233   x^2+4x^3+7x^4+3x^5+x^6,
%e A098233   x^2+13x^3+46x^4+47x^5+25x^6+6x^7+x^8,
%e A098233   x^2+40x^3+295x^4+587x^5+516x^6+235x^7+65x^8+10x^9+x^10,
%e A098233   ...
%e A098233 Triangle starts:
%e A098233   1;
%e A098233   1;
%e A098233   1,  1,   1;
%e A098233   1,  4,   7,   3,   1;
%e A098233   1, 13,  46,  47,  25,   6,  1;
%e A098233   1, 40, 295, 587, 516, 235, 65, 10, 1;
%e A098233   ...
%Y A098233 Cf. A360037, A360038, A360039, A020554 (row sums).
%K A098233 nonn,tabf
%O A098233 0,7
%A A098233 _N. J. A. Sloane_, Oct 26 2004