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A038164 Expansion of 1/((1-x)*(1-x^2))^4. 3

%I #22 Jan 13 2024 13:07:22

%S 1,4,14,36,85,176,344,624,1086,1800,2892,4488,6798,10032,14520,20592,

%T 28743,39468,53482,71500,94523,123552,159952,205088,260780,328848,

%U 411672,511632,631788,775200,945744,1147296,1384701,1662804

%N Expansion of 1/((1-x)*(1-x^2))^4.

%C From _Gary W. Adamson_, Mar 02 2010: (Start)

%C Given the tetrahedral numbers, A000292, shift the offset to 0; then

%C (1 + 4x + 10x^2 + 20x^3 + ...)*(1 + 4x^2 + 10x^4 + 20x^6 + ...) =

%C (1 + 4x^2 + 14x^3 + 36x^4 + ...) (End)

%H M. R. Bremner, <a href="http://arxiv.org/abs/1303.0920">Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures</a>, arXiv preprint arXiv:1303.0920 [math.RA], 2013.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).

%F a(2*k) = (4*k^2 + 24*k + 21)*binomial(k + 5, 5)/21 = A059600(k); a(2*k + 1) = 4*binomial(k + 6, 6)*(7 + 2*k)/7 = 4*A050486(k), k >= 0.

%F a(0)=1, a(1)=4, a(2)=14, a(3)=36, a(4)=85, a(5)=176, a(6)=344, a(7)=624, a(8)=1086, a(9)=1800, a(10)=2892, a(11)=4488, 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_, Jul 02 2011

%t CoefficientList[Series[1/((1-x)(1-x^2))^4,{x,0,40}],x] (* or *) LinearRecurrence[ {4,-2,-12,17,8,-28,8,17,-12,-2,4,-1},{1,4,14,36,85,176,344,624,1086,1800,2892,4488},40] (* _Harvey P. Dale_, Jul 02 2011 *)

%o (PARI) Vec(1/((1-x)*(1-x^2))^4 + O(x^40)) \\ _Michel Marcus_, Jan 13 2024

%Y Cf. A008619, A006918, A038163.

%Y Cf. A000292. - _Gary W. Adamson_, Mar 02 2010

%K nonn

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 28 03:28 EDT 2024. Contains 371235 sequences. (Running on oeis4.)