A101092 Second partial sums of fifth powers (A000584).

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

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Luciano Ancora, Recurrence relation for the second partial sums of m-th powers

Luciano Ancora, Second partial sums of the m-th powers

C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - N. J. A. Sloane, Mar 23 2014 (But beware of a typo)

Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]

Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

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

a(n) = 2*a(n-1) - a(n-2) + n^5. - Luciano Ancora, Jan 08 2015

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

Maple

f:=n->(2*n^7-7*n^5+7*n^3-2*n)/84;
[seq(f(n), n=0..50)];  # N. J. A. Sloane, Mar 23 2014

Mathematica

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 *)

Prog

(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
(PARI) a(n)=n*(n+1)*(n+2)*(n*(n+2)-1)*(2*n*(2 + n)+1)/84 \\ Charles R Greathouse IV, Apr 21 2015

Crossrefs

Cf. A000539, A101099.

Cf. A254640.

Sequence in context: A230803 A209808 A160146 * A233061 A281805 A034978

Adjacent sequences: A101089 A101090 A101091 * A101093 A101094 A101095

Keyword

nonn,easy,changed

Author

Cecilia Rossiter, Dec 14 2004

Extensions

Edited by Ralf Stephan, Dec 16 2004

Status

approved