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A162668
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a(n) = n*(n+1)*(n+2)*(n+3)/3.
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3
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0, 8, 40, 120, 280, 560, 1008, 1680, 2640, 3960, 5720, 8008, 10920, 14560, 19040, 24480, 31008, 38760, 47880, 58520, 70840, 85008, 101200, 119600, 140400, 163800, 190008, 219240, 251720, 287680, 327360, 371008, 418880, 471240, 528360, 590520
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OFFSET
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0,2
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COMMENTS
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a(n+3) is the number of equivalence classes of n-tuples from the set {1,0,-1} where the number of nonzero elements is 4 and two n-tuples are equivalent if they are negatives of each other. - Michael Somos, Oct 19 2022
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LINKS
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FORMULA
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G.f.: 8*x/(1-x)^5. (End)
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} sin(x)^7 * cos(x)^(2*n-1) dx). - Francesco Daddi, Aug 02 2011
E.g.f.: x*(24 + 36*x + 12*x^2 + x^3)*exp(x)/3. - G. C. Greubel, Aug 27 2019
Sum_{n>=1} 1/a(n) = 1/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2) - 8/3.
Product_{n>=1} 1-1/a(n) = 4*cos(sqrt(13)*Pi/2)*cosh(sqrt(3)*Pi/2)/(3*Pi^2). (End)
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EXAMPLE
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G.f. = 8*x + 40*x^2 + 120*x^3 + 280*x^4 + 560*x^5 + ... - Michael Somos, Oct 19 2022
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MAPLE
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MATHEMATICA
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PROG
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(Magma) [n*(n+1)*(n+2)*(n+3)/3: n in [0..40] ];
(Sage) [8*binomial(n+3, 4) for n in (0..40)] # G. C. Greubel, Aug 27 2019
(GAP) List([0..40], n-> 8*Binomial(n+3, 4)); # G. C. Greubel, Aug 27 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Definition factorized, offset corrected by R. J. Mathar, Jul 13 2009
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STATUS
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approved
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