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A145217
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a(n) is the self-convolution series of the sum of 4th powers of the first n natural numbers.
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2
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1, 32, 418, 3104, 16003, 64064, 213060, 614976, 1587333, 3742816, 8190182, 16832608, 32795399, 61021312, 109078664, 188234880, 314856201, 512202912, 812698666, 1260762272, 1916300683, 2858972864, 4193345740, 6055075520
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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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).
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FORMULA
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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]
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EXAMPLE
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a(3) = 418 because 1(3^4)+(2^4)(2^4)+(3^4)1= 418
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MAPLE
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f:=n->(n^9+20*n^3-21*n)/630;
[seq(f(n), n=0..50)]; # N. J. A. Sloane, Mar 23 2014
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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a(n) = Conv(A000538, A000538).
Sequence in context: A061594 A145403 A125116 * A125444 A022692 A160142
Adjacent sequences: A145214 A145215 A145216 * A145218 A145219 A145220
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KEYWORD
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nonn,easy
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AUTHOR
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Abdullahi Umar, Oct 05 2008
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STATUS
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approved
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