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

1,2

Comments

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

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.

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]

Example

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

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 *)

Prog

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

Crossrefs

Cf. A098360.

Sequence in context: A279163 A101377 A302897 * A200173 A226761 A250169

Adjacent sequences: A145213 A145214 A145215 * A145217 A145218 A145219

Keyword

nonn

Author

Abdullahi Umar, Oct 05 2008

Extensions

Name corrected by Clark Kimberling, Jun 17 2012

Status

approved