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A210569
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a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30.
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6
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0, 0, 0, 0, 4, 24, 84, 224, 504, 1008, 1848, 3168, 5148, 8008, 12012, 17472, 24752, 34272, 46512, 62016, 81396, 105336, 134596, 170016, 212520, 263120, 322920, 393120, 475020, 570024, 679644, 805504, 949344, 1113024, 1298528, 1507968, 1743588, 2007768, 2303028
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OFFSET
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0,5
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COMMENTS
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The following sequences are provided by the formula n*binomial(n,k) - binomial(n,k+1) = k*binomial(n+1,k+1):
. a(n) for k=4,
Sum of reciprocals of a(n), for n>3: 5/16.
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LINKS
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FORMULA
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G.f.: 4*x^4/(1-x)^6.
a(n) = n*binomial(n,4)-binomial(n,5) = 4*binomial(n+1,5) = 4*A000389(n+1).
(n-4)*a(n) = (n+1)*a(n-1).
Sum_{n>=4} (-1)^n/a(n) = 20*log(2) - 655/48. - Amiram Eldar, Jun 02 2022
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MAPLE
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f:=n->(n^5-5*n^3+4*n)/30;
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MATHEMATICA
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LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 0, 0, 4, 24}, 39]
CoefficientList[Series[4x^4/(1-x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Times@@@Partition[Range[-3, 40], 5, 1]/30 (* Harvey P. Dale, Sep 19 2020 *)
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PROG
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(Magma) [4*Binomial(n+1, 5): n in [0..38]];
(Maxima) makelist(coeff(taylor(4*x^4/(1-x)^6, x, 0, n), x, n), n, 0, 38);
(SageMath) [4*binomial(n+1, 5) for n in (0..40)] # G. C. Greubel, May 23 2022
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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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STATUS
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approved
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