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Expansion of (1-x)^(-3) * (1-x^2)^(-2).
(Formerly M2723 N1091)
16

%I M2723 N1091 #126 May 04 2023 17:33:24

%S 1,3,8,16,30,50,80,120,175,245,336,448,588,756,960,1200,1485,1815,

%T 2200,2640,3146,3718,4368,5096,5915,6825,7840,8960,10200,11560,13056,

%U 14688,16473,18411,20520,22800,25270,27930,30800,33880,37191,40733,44528

%N Expansion of (1-x)^(-3) * (1-x^2)^(-2).

%C Given an irregular triangular matrix M with the triangular numbers in every column shifted down twice for columns >0, A002624 = M * [1, 2, 3, ...]. Example: row 4 of triangle M = (15, 6, 1), then (15, 6, 1) dot (1, 2, 3) = a(4) = 30 = (15 + 12 + 3). - _Gary W. Adamson_, Mar 02 2010

%C The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums of A139600 are related to the sequence given above, e.g., Ze2(n) = a(n-1) - a(n-2) - a(n-3) + 4*a(n-4), with a(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - _Johannes W. Meijer_, Apr 29 2011

%C 8*a(n) + 16*a(n+1) + 16*a(n+2) is the number of ways to place 3 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly twice. Also true for the nonexistent terms for n=-1, n=-2 and n=-3 assuming that they are zeros. In graph-theory representation they thus form the corresponding open walk (Eulerian trail) with V={1,2,3} vertices and length of 2. - _Antal Pinter_, Dec 31 2015

%C a(n) is the number of partitions of n into parts with three kinds of 1 and two kinds of 2. - _Joerg Arndt_, Jan 18 2016

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002624/b002624.txt">Table of n, a(n) for n = 0..10000</a>

%H Steven Edwards and William Griffiths, <a href="https://www.fq.math.ca/Papers1/55-4/EdwardsGriffiths82617.pdf">Generalizations of Delannoy and cross polytope numbers</a>, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.

%H E. Fix and J. L. Hodges, Jr., <a href="http://www.jstor.org/stable/2236885">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312.

%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]

%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=204">Encyclopedia of Combinatorial Structures 204</a>

%H Antal Pinter, <a href="http://pinter.netii.net/queens_down.php?sifart=101&amp;nazart=K3Queens_en_rev2.pdf">Numerical solution of the k=3 Queens problem</a>, 2011, Q(n) at p.8.

%H Antal Pinter, <a href="http://pinter.netii.net/comb.htm#backtrack_v7pro">Software utility for enumerating positions of non-attacking and attacking chess pieces</a> , Backtrack_V7Pro

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).

%F a(n-1) = ( n^4 +10*n^3 +32*n^2 +32*n +(6*n +15)*(n mod 2) )/96.

%F From _Antal Pinter_, Oct 03 2014: (Start)

%F a(n) = C(n + 2, 2) + 2*C(n, 2) + 3*C(n - 2, 2) + 4*C(n - 4, 2) + ...

%F a(n) = Sum_{i = 1..z} i*C(n + 4 - 2i, 2) where z = (2*n + 3 + (-1)^n)/4.

%F a(n) = (3*(2*n + 7)*(-1)^n + 2*n^4 + 28*n^3 + 136*n^2 + 266*n + 171)/192.

%F (End)

%F a(n) = A007009(n+1) - A001752(n-1) for n>0. - _Antal Pinter_, Dec 27 2015

%F a(n) = Sum_{j=0..n+1} A006918(j). - _Richard Turk_, Feb 18 2016

%p A002624:=-1/(z+1)**2/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation

%t f[n_] := Block[{m = n - 1}, (m^4 + 10m^3 + 32m^2 + 32m + (6m + 15)Mod[m, 2])/96]; Table[ f[n], {n, 2, 45}]

%t (* Or *) CoefficientList[ Series[1/((1 - x)^3 (1 - x^2)^2), {x, 0, 44}], x] (* _Robert G. Wilson v_, Feb 26 2005 *)

%o (Magma) [( (n+1)^4 +10*(n+1)^3 +32*(n+1)^2 +32*(n+1) +(6*(n+1) +15)*((n+1) mod 2) )/96 : n in [0..50]]; // _Vincenzo Librandi_, Oct 08 2011

%o (PARI) Vec(1/(1-x)^3/(1-x^2)^2+O(x^99)) \\ _Charles R Greathouse IV_, Apr 19 2012

%o (PARI) a(n)=(n^4 + 14*n^3 + 68*n^2 + 136*n - n%2*(6*n + 21))/96 + 1 \\ _Charles R Greathouse IV_, Feb 18 2016

%Y Cf. A047659, A139600, A180662.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Formula and more terms from _Frank Ellermann_, Mar 14 2002