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 A059338 a(n) = Sum_{k=1..n} k^5 * binomial(n,k). 4
 1, 34, 342, 2192, 11000, 47232, 181888, 646144, 2156544, 6848000, 20877824, 61526016, 176171008, 492126208, 1345536000, 3610247168, 9526771712, 24769069056, 63546720256, 161087488000, 403925630976, 1002841309184 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES Finding a closed form for the sum was Problem 541 in the Fall 2000 issue of The Pentagon (official journal of Kappa Mu Epsilon). LINKS Harry J. Smith, Table of n, a(n) for n = 1..200 Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64). FORMULA The closed form comes from starting with (1+x)^n and repeatedly differentiating and multiplying by x. After five differentiations, substitute x=1 and get a(n) = 2^(n-5)*n^2*(n^3+10n^2+15n-10). G.f.: x*(16*x^4-32*x^3-6*x^2+22*x+1)/(2*x-1)^6. - Colin Barker, Sep 20 2012 MAPLE with(combinat): for n from 1 to 70 do printf(`%d, `, sum(k^5*binomial(n, k), k=1..n)) od: MATHEMATICA Table[Sum[k^5*Binomial[n, k], {k, 1, n}], {n, 1, 5}] (* or *) LinearRecurrence[{12, -60, 160, -240, 192, -64}, {1, 34, 342, 2192,   11000, 47232}, 10] (* G. C. Greubel, Jan 07 2017 *) PROG (PARI) { for (n = 1, 200, write("b059338.txt", n, " ", sum(k=1, n, k^5*binomial(n, k))); ) } \\ Harry J. Smith, Jun 26 2009 (PARI) Vec(x*(16*x^4-32*x^3-6*x^2+22*x+1)/(2*x-1)^6 + O(x^25)) \\ G. C. Greubel, Jan 07 2017 CROSSREFS Binomial transform of A000584. Sequence in context: A281805 A034978 A251938 * A301954 A244881 A296833 Adjacent sequences:  A059335 A059336 A059337 * A059339 A059340 A059341 KEYWORD nonn,easy AUTHOR Pat Costello (matcostello(AT)acs.eku.edu), Jan 26 2001 EXTENSIONS More terms from James A. Sellers, Jan 29 2001 STATUS approved

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)