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A145217
a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.
2
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
1,2
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
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).
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]
EXAMPLE
a(3) = 418 because 1(3^4)+(2^4)(2^4)+(3^4)1= 418
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 *)
PROG
(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
CROSSREFS
a(n) = Conv(A000538, A000538).
Sequence in context: A061594 A145403 A125116 * A125444 A022692 A160142
KEYWORD
nonn,easy
AUTHOR
Abdullahi Umar, Oct 05 2008
STATUS
approved