cp's OEIS Frontend

A145216 Self-convolution of (1^3, 2^3, 3^3, 4^3, ... ).

%I A145216 #36 Sep 07 2024 19:53:20
%S A145216 1,16,118,560,2003,5888,14988,34176,71445,139216,255970,448240,752999,
%T A145216 1220480,1917464,2931072,4373097,6384912,9142990,12865072,17817019,
%U A145216 24320384,32760740,43596800,57370365,74717136,96378426,123213808
%N A145216 Self-convolution of (1^3, 2^3, 3^3, 4^3, ... ).
%C A145216 Row sums of A098360. - _N. J. A. Sloane_, May 31 2009
%D A145216 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.
%H A145216 Vincenzo Librandi, <a href="/A145216/b145216.txt">Table of n, a(n) for n = 1..1000</a>
%H A145216 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 A145216 A. Umar, B. Yushau and B. M. Ghandi, <a href="https://www.researchgate.net/publication/261685299_Convolution_of_two_series">Convolution of two series</a>, Australian Senior Maths Journal 21(2) (2007), 6-11.
%F A145216 a(n) = C(n+2,3)*(3*n*(n+2)*(n^2+2*n+3)+16)/70.
%F A145216 G.f.: x*(1+4*x+x^2)^2/(1-x)^8. [_Joerg Arndt_, Jun 18 2012]
%e A145216 a(3) = 118 because 1*(3^3) + (2^3)*(2^3) + (3^3)*1 = 118.
%p A145216 f:=n->(3*n^7+7*n^3-10*n)/420;
%p A145216 [seq(f(n),n=0..50)];  # _N. J. A. Sloane_, Mar 23 2014
%t A145216 Table[Sum[(k - i)^3 (1 + i)^3, {i, 0, k - 1}], {k, 1, 35}] (* _Clark Kimberling_, Jun 17 2012 *)
%t A145216 CoefficientList[Series[(1 + 4 x + x^2)^2/(1 - x)^8, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)
%t A145216 Table[ListConvolve[Range[n]^3,Range[n]^3],{n,30}]//Flatten (* _Harvey P. Dale_, Sep 07 2024 *)
%o A145216 (Magma) [(3*n^7+7*n^3-10*n)/420: n in [2..40]]; // _Vincenzo Librandi_, Mar 24 2014
%Y A145216 Cf. A098360.
%K A145216 nonn
%O A145216 1,2
%A A145216 _Abdullahi Umar_, Oct 05 2008
%E A145216 Name corrected by _Clark Kimberling_, Jun 17 2012