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A069038
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G.f.: x*(1+x)^4/(1-x)^6.
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8
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0, 1, 10, 51, 180, 501, 1182, 2471, 4712, 8361, 14002, 22363, 34332, 50973, 73542, 103503, 142544, 192593, 255834, 334723, 432004, 550725, 694254, 866295, 1070904, 1312505, 1595906, 1926315, 2309356, 2751085, 3258006, 3837087, 4495776
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Figurate numbers based on the 5-dimensional regular convex polytope, variously called the 5-dimensional hyperoctahedron, or the 5-dimensional cross-polytope, which is represented by the Schlaefli symbol {3, 3, 3, 4}. It is the dual of the 5-dimensional hypercube. Hyun Kwang Kim asserts that every nonnegative integer can be represented by the sum of no more than 14 of these 5-crosspolytope numbers. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 16 2004
If Y_i (i=1,2,3,4) are 2-blocks of a (n+4)-set X then a(n-4) is the number of 9-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan R. Janjic (agnus(AT)blic.net), Oct 28 2007
Starting with 1 = binomial transform of [1, 9, 32, 56, 48, 16, 0, 0, 0,...], where (1, 9, 32, 56, 48, 16) = row 5 of the Chebyshev triangle A081277. Also = row 5 of the array in A142978. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008
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REFERENCES
| H. S. M. Coxeter, Regular Polytopes, New York: Dover, 1973.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to linear recurrences with constant coefficients
Milan Janjic, Two Enumerative Functions
H. K. Kim, On Regular polytope numbers.
J. V. Post, Table of polytope numbers, Sorted, Through 1,000,000.
J. V. Post, Math Pages.
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FORMULA
| Recurrence: a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6).
a(n) = 5-crosspolytope(n) = n*(2*n^4 + 10*n^2 + 3)/15. E.g. a(5) = 501 because 5*(2*5^4 + 10*5^2 + 3)/15 = 501. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 16 2004
a(n) = C(n+4,5) + 4 C(n+3,5) + 6 C(n+2,5) + 4 C(n+1,5) + C(n,5)
sum(1/(((1/15)*n*(2*n^4+10*n^2+3)*n!)),n=1..infinity)=hypergeom([1, 1, 1+I*sqrt(10-2*sqrt(19))*(1/2), 1-I*sqrt(10-2*sqrt(19))*(1/2), 1+I*sqrt(10+2*sqrt(19))*(1/2), 1-I*sqrt(10+2*sqrt(19))*(1/2)], [2, 2, 2+I*sqrt(10-2*sqrt(19))*(1/2), 2-I*sqrt(10-2*sqrt(19))*(1/2), 2+I*sqrt(10+2*sqrt(19))*(1/2), 2-I*sqrt(10+2*sqrt(19))*(1/2)], 1)=1.05351734968093116819345664995829700099916... [From Stephen Crowley (crow(AT)crowlogic.net), Jul 14 2009]
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MAPLE
| al:=proc(s, n) binomial(n+s-1, s); end; be:=proc(d, n) local r; add( (-1)^r*binomial(d-1, r)*2^(d-1-r)*al(d-r, n), r=0..d-1); end; [seq(be(5, n), n=0..100)];
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PROG
| (MAGMA) [n*(2*n^4 + 10*n^2 + 3)/15: n in [0..40]]; // Vincenzo Librandi, May 22 2011
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CROSSREFS
| Cf. A000332, A014820, A005900.
Cf. A142978, A081277.
Sequence in context: A143855 A124162 A077044 * A030183 A135242 A041186
Adjacent sequences: A069035 A069036 A069037 * A069039 A069040 A069041
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KEYWORD
| nonn
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 03 2002
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