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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 (list; graph; refs; listen; history; text; internal format)
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

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).

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

Adjacent sequences:  A145214 A145215 A145216 * A145218 A145219 A145220

KEYWORD

nonn,easy

AUTHOR

Abdullahi Umar, Oct 05 2008

STATUS

approved

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Last modified July 6 14:10 EDT 2022. Contains 355110 sequences. (Running on oeis4.)