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%I #98 Oct 24 2023 10:09:38
%S 0,1,4,13,38,104,272,688,1696,4096,9728,22784,52736,120832,274432,
%T 618496,1384448,3080192,6815744,15007744,32899072,71827456,156237824,
%U 338690048,731906048,1577058304,3388997632,7264534528,15535702016
%N a(n) = T(n,2), array T as in A049600.
%C Refer to A089378 and A075729 for the definition of hierarchies, subhierarchies and one-step transitions. - _Thomas Wieder_, Feb 28 2004
%C 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 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 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 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 Vincenzo Librandi, <a href="/A049611/b049611.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H 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 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 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 S. Kitaev, J. Remmel, <a href="http://arxiv.org/abs/1503.00914">p-Ascent Sequences</a>, arXiv:1503.00914 [math.CO], 2015.
%H 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 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 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F G.f.: x*(1-x)^2/(1-2*x)^3.
%F 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 a(n) = 2^(n-4)*(n^2+5*n+2) - 0^n/8. - _Paul Barry_, Jun 09 2003
%F a(n+2) = A001787(n+2) + A001788(n). - _Creighton Dement_, Aug 02 2005
%F a(n) = Hyper2F1([-n+1, 3], [1], -1) for n>0. - _Peter Luschny_, Aug 02 2014
%F 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 CoefficientList[Series[x (1-x)^2/(1-2x)^3,{x,0,40}],x] (* _Harvey P. Dale_, Sep 24 2013 *)
%o (PARI) concat(0, Vec(x*(1-x)^2/(1-2*x)^3+O(x^99))) \\ _Charles R Greathouse IV_, Jun 12 2015
%Y a(n+1)= A055252(n, 0), n >= 0. Row sums of triangle A055249.
%Y Cf. A001793, A058396, A075729, A089378, A000217.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_
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