A145218 a(n) is the self-convolution series of the sum of 5th powers of the first n natural numbers.

1, 64, 1510, 17600, 130835, 713216, 3098604, 11320320, 36074325, 102925120, 268038706, 646519744, 1460878055, 3120396800, 6346379480, 12363588096, 23184837609, 42023883840, 73881649150, 126362703040, 210792998011, 343726413824, 548946959300, 860095808000

Offset

1,2

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.

A. Umar, B. Yushau and B. M. Ghandi, "Convolution of two series", Australian Senior Maths. Journal, 21(2) (2007), 6-11.

Links

T. D. Noe, 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

Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Formula

a(n) = C(n+2, 3)*(n^8 + 8*n^7 + 29*n^6 + 62*n^5 + 86*n^4 + 80*n^3 + 28*n^2 - 24*n + 192)/462.

G.f.: x*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1)^2/(x-1)^12. [Colin Barker, Jul 08 2012]

Example

a(3) = 1510 because 1(3^5)+(2^5)(2^5)+(3^5)1= 1510

Maple

f:=n->(n^11-22*n^5+231*n^3-210*n)/2772;
[seq(f(n), n=0..50)]; # N. J. A. Sloane, Mar 23 2014

Mathematica

CoefficientList[Series[(x^4 + 26 x^3 + 66 x^2 + 26 x + 1)^2/(x - 1)^12, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

Prog

(Magma) [Binomial(n+2, 3)*(n^8+8*n^7+29*n^6+62*n^5+86*n^4 +80*n^3+28*n^2-24*n+192)/462: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

Crossrefs

a(n)=Conv(A000539, A000539)

Sequence in context: A239442 A240930 A208313 * A282526 A014794 A224282

Adjacent sequences: A145215 A145216 A145217 * A145219 A145220 A145221

Keyword

nonn,easy,changed

Author

Abdullahi Umar, Oct 05 2008

Status

approved