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A145217 a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.

%I A145217 #34 Apr 14 2024 03:47:02
%S A145217 1,32,418,3104,16003,64064,213060,614976,1587333,3742816,8190182,
%T A145217 16832608,32795399,61021312,109078664,188234880,314856201,512202912,
%U A145217 812698666,1260762272,1916300683,2858972864,4193345740,6055075520
%N A145217 a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.
%D A145217 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 A145217 A. Umar, B. Yushau and B. M. Ghandi, "Convolution of two series", Australian Senior Maths. Journal, 21(2) (2007), 6-11.
%H A145217 Vincenzo Librandi, <a href="/A145217/b145217.txt">Table of n, a(n) for n = 1..1000</a>
%H A145217 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 A145217 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A145217 a(n) = C(n+2,3)*(n*(n+2)*(n^4+4*n^3+8*n^2+8*n+6)+24)/105.
%F A145217 G.f.: x*(1+x)^2*(1+10*x+x^2)^2/(1-x)^10. [_Colin Barker_, May 25 2012]
%e A145217 a(3) = 418 because 1(3^4)+(2^4)(2^4)+(3^4)1= 418
%p A145217 f:=n->(n^9+20*n^3-21*n)/630;
%p A145217 [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014
%t A145217 CoefficientList[Series[(1 + x)^2 (1 + 10 x + x^2)^2/(1 - x)^10, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%t A145217 LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,32,418,3104,16003,64064,213060,614976,1587333,3742816},30] (* _Harvey P. Dale_, May 19 2021 *)
%o A145217 (Magma) [Binomial(n+2,3)*(n*(n+2)*(n^4+4*n^3+8*n^2+8*n+6)+24)/105: n in [1..40]]; // _Vincenzo Librandi_, Mar 24 2014
%Y A145217 a(n) = Conv(A000538, A000538).
%K A145217 nonn,easy
%O A145217 1,2
%A A145217 _Abdullahi Umar_, Oct 05 2008