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A145218 a(n) is the self-convolution series of the sum of 5th powers of the first n natural numbers.

%I A145218 #27 Apr 14 2024 03:47:07
%S A145218 1,64,1510,17600,130835,713216,3098604,11320320,36074325,102925120,
%T A145218 268038706,646519744,1460878055,3120396800,6346379480,12363588096,
%U A145218 23184837609,42023883840,73881649150,126362703040,210792998011,343726413824,548946959300,860095808000
%N A145218 a(n) is the self-convolution series of the sum of 5th powers of the first n natural numbers.
%D A145218 A. Umar, B. Yushau and B. M. Ghandi, (2006), "Patterns in convolution of two series", in Stewart, S. M., Olearski, J. E. and Thompson, D. (Eds), Proceedings of the Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (pp. 95-101). METSMaC: Abu Dhabi.
%D A145218 A. Umar, B. Yushau and B. M. Ghandi, "Convolution of two series", Australian Senior Maths. Journal, 21(2) (2007), 6-11.
%H A145218 T. D. Noe, <a href="/A145218/b145218.txt">Table of n, a(n) for n = 1..1000</a>
%H A145218 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 2. - _N. J. A. Sloane_, Mar 23 2014
%H A145218 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F A145218 a(n) = C(n+2, 3)*(n^8 + 8*n^7 + 29*n^6 + 62*n^5 + 86*n^4 + 80*n^3 + 28*n^2 - 24*n + 192)/462.
%F A145218 G.f.: x*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1)^2/(x-1)^12. [_Colin Barker_, Jul 08 2012]
%e A145218 a(3) = 1510 because 1(3^5)+(2^5)(2^5)+(3^5)1= 1510
%p A145218 f:=n->(n^11-22*n^5+231*n^3-210*n)/2772;
%p A145218 [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014
%t A145218 CoefficientList[Series[(x^4 + 26 x^3 + 66 x^2 + 26 x + 1)^2/(x - 1)^12, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%o A145218 (Magma) [Binomial(n+2,3)*(n^8+8*n^7+29*n^6+62*n^5+86*n^4 +80*n^3+28*n^2-24*n+192)/462: n in [1..40]]; // _Vincenzo Librandi_, Mar 24 2014
%Y A145218 a(n)=Conv(A000539, A000539)
%K A145218 nonn,easy
%O A145218 1,2
%A A145218 _Abdullahi Umar_, Oct 05 2008