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Second partial sums of fifth powers (A000584).
12

%I #65 Oct 04 2024 06:52:55

%S 1,34,310,1610,6035,18236,47244,109020,229845,450670,832546,1463254,

%T 2465255,4005080,6304280,9652056,14419689,21076890,30210190,42543490,

%U 58960891,80531924,108539300,144509300,190244925,247861926,319827834,409004110,518691535

%N Second partial sums of fifth powers (A000584).

%H Vincenzo Librandi, <a href="/A101092/b101092.txt">Table of n, a(n) for n = 1..1000</a>

%H Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>

%H Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>

%H C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, 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)

%H Cecilia Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Dead link]

%H Cecilia Rossiter, <a href="/A101092/a101092_1.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a> [Cached copy, May 15 2013]

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = (n*(1 + n)*(2 + n)*(-1 + n*(2 + n))*(1 + 2*n*(2 + n)))/84.

%F G.f.: x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^8. - _Colin Barker_, Apr 16 2012

%F a(n) = Sum_{i=1..n} i*(n+1-i)^5, by the definition. - _Bruno Berselli_, Jan 31 2014

%F a(n) = 2*a(n-1) - a(n-2) + n^5. - _Luciano Ancora_, Jan 08 2015

%F 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

%p f:=n->(2*n^7-7*n^5+7*n^3-2*n)/84;

%p [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014

%t CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)

%t 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 *)

%o (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

%o (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

%o (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

%o (Python)

%o def A101092(n): return n*(n*(n*(n*(n*(n*(n+7<<1)+35)+35)+7)-7)-2)//84 # _Chai Wah Wu_, Oct 03 2024

%Y Cf. A000539, A101099.

%Y Cf. A254640.

%K nonn,easy

%O 1,2

%A Cecilia Rossiter, Dec 14 2004

%E Edited by _Ralf Stephan_, Dec 16 2004