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A145216
Self-convolution of (1^3, 2^3, 3^3, 4^3, ... ).
5
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
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 *)
Table[ListConvolve[Range[n]^3, Range[n]^3], {n, 30}]//Flatten (* Harvey P. Dale, Sep 07 2024 *)
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
KEYWORD
nonn
AUTHOR
Abdullahi Umar, Oct 05 2008
EXTENSIONS
Name corrected by Clark Kimberling, Jun 17 2012
STATUS
approved