A031164 Irreducible Euler sums of weight 8 and depth 10+2n.
1, 4, 15, 40, 99, 212, 429, 800, 1430, 2424, 3978, 6288, 9690, 14520, 21318, 30624, 43263, 60060, 82225, 110968, 148005, 195052, 254475, 328640, 420732, 533936, 672452, 840480, 1043460, 1286832, 1577532, 1922496, 2330445
Offset: 0
Links
- David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
- David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
- Index entries for sequences related to Lyndon words
- Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).
Programs
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Mathematica
Table[(Binomial[n+8,7]-If[OddQ[n],1,0]Binomial[(n+7)/2,3])/8,{n,0,40}] (* or *) CoefficientList[Series[(1+x^2)/((1-x)^8 (1+x)^4),{x,0,40}],x] (* Harvey P. Dale, Jun 20 2011 *) -
PARI
A031164(n)=(binomial(n+8,7)-if(n%2,binomial(n\2+4,3)))>>3 \\ M. F. Hasler, May 02 2009
Formula
G.f.: (1+x^2)/((1-x)*(1-x^2))^4
a(n) = [C(n+8,7)-(n%2)*C((n+7)/2,3)]/8, where C = binomial, n%2 = parity of n (=1 if odd, 0 else). - M. F. Hasler, May 02 2009
a(0)=1, a(1)=4, a(2)=15, a(3)=40, a(4)=99, a(5)=212, a(6)=429, a(7)=800, a(8)=1430, a(9)=2424, a(10)=3978, a(11)=6288, a(n) = 4*a(n-1)-2*a(n-2)-12*a(n-3)+17*a(n-4)+8*a(n-5)-28*a(n-6)+8*a(n-7)+17*a(n-8)-12*a(n-9)- 2*a(n-10)+4*a(n-11)-a(n-12). - Harvey P. Dale, Jun 20 2011
G.f.: ((-1+x)^-8-(-1+x^2)^-4)/(8*x). - Herbert Kociemba, Oct 16 2016
Comments