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

1, 16, 118, 560, 2003, 5888, 14988, 34176, 71445, 139216, 255970, 448240, 752999, 1220480, 1917464, 2931072, 4373097, 6384912, 9142990, 12865072, 17817019, 24320384, 32760740, 43596800, 57370365, 74717136, 96378426, 123213808

Offset: 1

Author

Abdullahi Umar, Oct 05 2008

Keywords

• nonn

Comments

Row sums of A098360. - N. J. A. Sloane, May 31 2009

Examples

			a(3) = 118 because 1*(3^3) + (2^3)*(2^3) + (3^3)*1 = 118.
		

References

• 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• 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 2. - _N. J. A. Sloane_, Mar 23 2014
• A. Umar, B. Yushau and B. M. Ghandi, Convolution of two series, Australian Senior Maths Journal 21(2) (2007), 6-11.

Crossrefs

Cf. A098360.

Programs

Magma

[(3*n^7+7*n^3-10*n)/420: n in [2..40]]; // Vincenzo Librandi, Mar 24 2014

Maple

f:=n->(3*n^7+7*n^3-10*n)/420;
[seq(f(n),n=0..50)];  # N. J. A. Sloane, Mar 23 2014

Mathematica

Table[Sum[(k - i)^3 (1 + i)^3, {i, 0, k - 1}], {k, 1, 35}] (* Clark Kimberling, Jun 17 2012 *)
CoefficientList[Series[(1 + 4 x + x^2)^2/(1 - x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Table[ListConvolve[Range[n]^3,Range[n]^3],{n,30}]//Flatten (* Harvey P. Dale, Sep 07 2024 *)

Formula

a(n) = C(n+2,3)*(3*n*(n+2)*(n^2+2*n+3)+16)/70.

G.f.: x*(1+4*x+x^2)^2/(1-x)^8. [Joerg Arndt, Jun 18 2012]

Extensions

Name corrected by Clark Kimberling, Jun 17 2012