This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049612 #66 Jan 27 2024 10:29:25
%S A049612 0,1,5,19,63,192,552,1520,4048,10496,26624,66304,162560,393216,940032,
%T A049612 2224128,5214208,12124160,27983872,64159744,146210816,331350016,
%U A049612 747110400,1676673024,3746562048,8338276352,18488492032
%N A049612 a(n) = T(n,3), array T as in A049600.
%C A049612 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 A049612 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 A049612 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 A049612 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 A049612 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A049612 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 A049612 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 A049612 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16)
%F A049612 G.f.: x*(1-x)^3 /(1-2*x)^4.
%F A049612 a(n) = Sum_{k=0..floor((n+3)/2)} C(n+3, 2k)*C(k+1, 3). - _Paul Barry_, May 15 2003
%F A049612 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 A049612 Binomial transform of the tetrahedral numbers A000292 when omitting the initial 0 in both sequences. - _Carl Najafi_, Sep 08 2011
%t A049612 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 A049612 Cf. A049600.
%Y A049612 Row sums of triangle A055252. a(n+1) = A055584(n, 0), n >= 0.
%K A049612 nonn
%O A049612 0,3
%A A049612 _Clark Kimberling_