A145217 a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.

1, 32, 418, 3104, 16003, 64064, 213060, 614976, 1587333, 3742816, 8190182, 16832608, 32795399, 61021312, 109078664, 188234880, 314856201, 512202912, 812698666, 1260762272, 1916300683, 2858972864, 4193345740, 6055075520

Offset: 1

Author

Abdullahi Umar, Oct 05 2008

Keywords

• nonn
• easy

Examples

			a(3) = 418 because 1(3^4)+(2^4)(2^4)+(3^4)1= 418
		

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

• 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
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

a(n) = Conv(A000538, A000538).

Programs

Magma

[Binomial(n+2,3)*(n*(n+2)*(n^4+4*n^3+8*n^2+8*n+6)+24)/105: n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

Maple

f:=n->(n^9+20*n^3-21*n)/630;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

Mathematica

CoefficientList[Series[(1 + x)^2 (1 + 10 x + x^2)^2/(1 - x)^10, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,32,418,3104,16003,64064,213060,614976,1587333,3742816},30] (* Harvey P. Dale, May 19 2021 *)

Formula

a(n) = C(n+2,3)*(n*(n+2)*(n^4+4*n^3+8*n^2+8*n+6)+24)/105.

G.f.: x*(1+x)^2*(1+10*x+x^2)^2/(1-x)^10. [Colin Barker, May 25 2012]