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

1, 4, 14, 36, 85, 176, 344, 624, 1086, 1800, 2892, 4488, 6798, 10032, 14520, 20592, 28743, 39468, 53482, 71500, 94523, 123552, 159952, 205088, 260780, 328848, 411672, 511632, 631788, 775200, 945744, 1147296, 1384701, 1662804

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From Gary W. Adamson, Mar 02 2010: (Start)

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

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

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

Links

• M. R. Bremner, Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures, arXiv preprint arXiv:1303.0920 [math.RA], 2013.
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).

Crossrefs

Cf. A008619, A006918, A038163.

Cf. A000292. - Gary W. Adamson, Mar 02 2010

Programs

Mathematica

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 *)

PARI

Vec(1/((1-x)*(1-x^2))^4 + O(x^40)) \\ Michel Marcus, Jan 13 2024

Formula

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.

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