A002624 Expansion of (1-x)^(-3) * (1-x^2)^(-2). (Formerly M2723 N1091)

1, 3, 8, 16, 30, 50, 80, 120, 175, 245, 336, 448, 588, 756, 960, 1200, 1485, 1815, 2200, 2640, 3146, 3718, 4368, 5096, 5915, 6825, 7840, 8960, 10200, 11560, 13056, 14688, 16473, 18411, 20520, 22800, 25270, 27930, 30800, 33880, 37191, 40733, 44528

Offset

0,2

Comments

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

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

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

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

References

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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.

E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]

Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 204

Antal Pinter, Numerical solution of the k=3 Queens problem, 2011, Q(n) at p.8.

Antal Pinter, Software utility for enumerating positions of non-attacking and attacking chess pieces , Backtrack_V7Pro

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Formula

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

From Antal Pinter, Oct 03 2014: (Start)

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

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

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

(End)

a(n) = A007009(n+1) - A001752(n-1) for n>0. - Antal Pinter, Dec 27 2015

a(n) = Sum_{j=0..n+1} A006918(j). - Richard Turk, Feb 18 2016

Maple

A002624:=-1/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation

Mathematica

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

Prog

(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
(PARI) Vec(1/(1-x)^3/(1-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Apr 19 2012
(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

Crossrefs

Cf. A047659, A139600, A180662.

Sequence in context: A167616 A009439 A000233 * A293358 A227265 A295960

Adjacent sequences: A002621 A002622 A002623 * A002625 A002626 A002627

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Formula and more terms from Frank Ellermann, Mar 14 2002

Status

approved