A059338 a(n) = Sum_{k=1..n} k^5 * binomial(n,k).

1, 34, 342, 2192, 11000, 47232, 181888, 646144, 2156544, 6848000, 20877824, 61526016, 176171008, 492126208, 1345536000, 3610247168, 9526771712, 24769069056, 63546720256, 161087488000, 403925630976, 1002841309184

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 A368719 A362953

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