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%I A049611 #98 Oct 24 2023 10:09:38
%S A049611 0,1,4,13,38,104,272,688,1696,4096,9728,22784,52736,120832,274432,
%T A049611 618496,1384448,3080192,6815744,15007744,32899072,71827456,156237824,
%U A049611 338690048,731906048,1577058304,3388997632,7264534528,15535702016
%N A049611 a(n) = T(n,2), array T as in A049600.
%C A049611 Refer to A089378 and A075729 for the definition of hierarchies, subhierarchies and one-step transitions. - _Thomas Wieder_, Feb 28 2004
%C A049611 We may ask for the number of one-step transitions (NOOST) between all unlabeled hierarchies of n elements with the restriction that no subhierarchies are allowed. As an example, consider n = 4 and the hierarchy H1 = [[2,2]] with two elements on level 1 and two on level 2. Starting from H1 the hierarchies [[1, 3]], [[2, 1, 1]], [[1, 2, 1]] can be reached by moving one element only, but [[1, 1, 2]] cannot be reached in a one-step transitition. The solution is n = 1, NOOST = 0; n = 2, NOOST = 1; n = 3, NOOST = 4; n = 4, NOOST = 13; n = 5, NOOST = 38; n = 6, NOOST = 104; n = 7, NOOST = 272; n = 8, NOOST = 688; n = 9, NOOST = 1696. This is sequence A049611. - _Thomas Wieder_, Feb 28 2004
%C A049611 If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - _Milan Janjic_, Nov 18 2007
%C A049611 In each composition (ordered partition) of the integer n, circle the first summand once, circle the second summand twice, etc. a(n) is the total number of circles in all compositions of n (that is, add k*(k+1)/2 for each composition into k parts). Note the O.g.f. is B(A(x)) where A(x)= x/(1-x) and B(x)= x/(1-x)^3.
%C A049611 This is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the triangular number sequence A000217. See a Feb 17 2017 comment on A097805. - _Wolfdieter Lang_, Feb 17 2017
%H A049611 Vincenzo Librandi, <a href="/A049611/b049611.txt">Table of n, a(n) for n = 0..1000</a>
%H A049611 Robert Davis, Greg Simay, <a href="https://arxiv.org/abs/2001.11089">Further Combinatorics and Applications of Two-Toned Tilings</a>, arXiv:2001.11089 [math.CO], 2020.
%H A049611 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A049611 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.html">On a class of polynomials with integer coefficients</a>, JIS 11 (2008) 08.5.2.
%H A049611 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H A049611 M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H A049611 S. Kitaev, J. Remmel, <a href="http://arxiv.org/abs/1503.00914">p-Ascent Sequences</a>, arXiv:1503.00914 [math.CO], 2015.
%H A049611 Sergey Kitaev, J. B. 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 A049611 Igor Makhlin, <a href="https://arxiv.org/abs/2003.02916">Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties</a>, arXiv:2003.02916 [math.CO], 2020.
%H A049611 Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, <a href="http://dx.doi.org/10.17654/MS101081631">On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type</a>, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
%H A049611 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A049611 G.f.: x*(1-x)^2/(1-2*x)^3.
%F A049611 Binomial transform of quarter squares A002620(n+1): a(n) = Sum_{k=0..n} binomial(n, k)*floor((k+1)^2/4). - _Paul Barry_, May 27 2003
%F A049611 a(n) = 2^(n-4)*(n^2+5*n+2) - 0^n/8. - _Paul Barry_, Jun 09 2003
%F A049611 a(n+2) = A001787(n+2) + A001788(n). - _Creighton Dement_, Aug 02 2005
%F A049611 a(n) = Hyper2F1([-n+1, 3], [1], -1) for n>0. - _Peter Luschny_, Aug 02 2014
%F A049611 a(n) = Sum_{k=0..n-1} Sum_{j=0..n-1} Sum_{i=0..n-1} binomial(n-1, i+j+k). - _Yalcin Aktar_, Aug 27 2023
%t A049611 CoefficientList[Series[x (1-x)^2/(1-2x)^3,{x,0,40}],x] (* _Harvey P. Dale_, Sep 24 2013 *)
%o A049611 (PARI) concat(0, Vec(x*(1-x)^2/(1-2*x)^3+O(x^99))) \\ _Charles R Greathouse IV_, Jun 12 2015
%Y A049611 a(n+1)= A055252(n, 0), n >= 0. Row sums of triangle A055249.
%Y A049611 Cf. A001793, A058396, A075729, A089378, A000217.
%K A049611 nonn,easy
%O A049611 0,3
%A A049611 _Clark Kimberling_

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