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A101092
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Second partial sums of fifth powers (A000584).
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12
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1, 34, 310, 1610, 6035, 18236, 47244, 109020, 229845, 450670, 832546, 1463254, 2465255, 4005080, 6304280, 9652056, 14419689, 21076890, 30210190, 42543490, 58960891, 80531924, 108539300, 144509300, 190244925, 247861926, 319827834, 409004110, 518691535
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (n*(1 + n)*(2 + n)*(-1 + n*(2 + n))*(1 + 2*n*(2 + n)))/84.
G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^8. - Colin Barker, Apr 16 2012
a(n) = Sum_{i=1..n} i*(n+1-i)^5, by the definition. - Bruno Berselli, Jan 31 2014
E.g.f.: exp(x)*x*(84 + 1344*x + 2954*x^2 + 1995*x^3 + 525*x^4 + 56*x^5 + 2*x^6)/84. - Stefano Spezia, May 04 2024
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MAPLE
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f:=n->(2*n^7-7*n^5+7*n^3-2*n)/84;
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MATHEMATICA
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CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Nest[Accumulate, Range[30]^5, 2] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 34, 310, 1610, 6035, 18236, 47244, 109020}, 30] (* Harvey P. Dale, Aug 22 2022 *)
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PROG
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(Magma) [(n*(1+n)*(2+n)*(-1+n*(2+n))*(1+2*n*(2+n)))/84: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014
(Sage) [n*(1+n)*(2+n)*(n*(2+n)-1)*(1+2*n*(2+n))/84 for n in range(1, 30)] # Danny Rorabaugh, Apr 21 2015
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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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Cecilia Rossiter, Dec 14 2004
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EXTENSIONS
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STATUS
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approved
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