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A002624
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Expansion of (1-x)^-3 (1-x^2 )^-2.
(Formerly M2723 N1091)
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9
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 02 2010: (Start)
Given an irregular triangular matrix M with the triangular series 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). (End)
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums of A139600 are related to the sequence given above, e.g. Ze2(n) = A002624(n-1) - A002624(n-2) - A002624(n-3) + 4*A002624(n-4), with A002624(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. [From Johannes W. Meijer, Apr 29 2011]
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REFERENCES
| E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
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).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 204
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).
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FORMULA
| a(n-1)= ( n^4 +10*n^3 +32*n^2 +32*n +(6*n +15)*(n mod 2) )/96.
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MAPLE
| A002624:=-1/(z+1)**2/(z-1)**5; [S. Plouffe in his 1992 dissertation.]
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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] (from Robert G. Wilson v, Feb 26 2005)
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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
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CROSSREFS
| Sequence in context: A167616 A009439 A000233 * A068039 A188123 A081661
Adjacent sequences: A002621 A002622 A002623 * A002625 A002626 A002627
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Formula and more terms from Frank.Ellermann(AT)t-online.de, Mar 14 2002
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