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%I #66 Jan 27 2024 10:29:25
%S 0,1,5,19,63,192,552,1520,4048,10496,26624,66304,162560,393216,940032,
%T 2224128,5214208,12124160,27983872,64159744,146210816,331350016,
%U 747110400,1676673024,3746562048,8338276352,18488492032
%N a(n) = T(n,3), array T as in A049600.
%C If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - _Milan Janjic_, Nov 18 2007
%D Robert Cori, Gabor Hetyei, Genus one partitions, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, pp. 333-344, <hal-01207612>
%H R. Cori and G. Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
%H R. Cori and G. Hetyei, <a href="https://doi.org/10.46298/dmtcs.2404">How to count genus one partitions</a>, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H S. Kitaev and J. Remmel, <a href="http://arxiv.org/abs/1503.00914">p-Ascent Sequences</a>, arXiv preprint arXiv:1503.00914 [math.CO], 2015.
%H Sergey Kitaev and Jeffrey Remmel, <a href="https://pure.strath.ac.uk/portal/files/46917816/Kitaev_Remmel_JC2016_a_note_on_p_ascent_sequences.pdf">A note on p-ascent sequences</a>, preprint, 2016.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16)
%F G.f.: x*(1-x)^3 /(1-2*x)^4.
%F a(n) = Sum_{k=0..floor((n+3)/2)} C(n+3, 2k)*C(k+1, 3). - _Paul Barry_, May 15 2003
%F a(n+1) = 2^n*n^3/48 + 5*2^n*n^2/16 + 7*2^n*n/6 + 2^n, n>=1. - _Milan Janjic_, Nov 18 2007
%F Binomial transform of the tetrahedral numbers A000292 when omitting the initial 0 in both sequences. - _Carl Najafi_, Sep 08 2011
%t CoefficientList[Series[x (1-x)^3/(1-2x)^4,{x,0,30}],x] (* or *) Join[ {0},LinearRecurrence[{8,-24,32,-16},{1,5,19,63},30]] (* _Harvey P. Dale_, Jan 07 2014 *)
%Y Cf. A049600.
%Y Row sums of triangle A055252. a(n+1) = A055584(n, 0), n >= 0.
%K nonn
%O 0,3
%A _Clark Kimberling_
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