A290989 Expansion of x^6*(1 + x^3)/(1 - 4*x + 5*x^2 - x^3 - 2*x^4 + x^6 + x^7 - 2*x^8 + x^9).

1, 4, 11, 26, 55, 109, 208, 389, 722, 1339, 2488, 4634, 8646, 16146, 30160, 56333, 105198, 196413, 366672, 684475, 1277701, 2385080, 4452277, 8311254, 15515091, 28963012, 54067156, 100930660, 188413624, 351723304, 656583197

Offset: 6

Author

R. J. Mathar, Aug 16 2017

Keywords

• nonn
• easy

Comments

This corresponds to S(213,1,x) of Langley if one uses Theorem 8. Note that all three expressions for S(213;t,x), S(213;1,x) and the series on page 22 are mutually incompatible, so we show the sequence one would most likely see in other publications.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 6..1000
• T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,0,-1,-1,2,-1).

Crossrefs

Cf. A059633, A099530, A290986, A290987.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^6*(1+x^3)/((1-x)*(1-2*x+x^3-x^4)*(1-x+x^4)) )); // G. C. Greubel, Apr 12 2023

Mathematica

DeleteCases[#, 0] &@ CoefficientList[Series[x^6*(1+x^3)/(1 -4x +5x^2 -x^3 -2x^4 +x^6 +x^7 -2x^8 +x^9), {x, 0, 36}], x] (* Michael De Vlieger, Aug 16 2017 *)
LinearRecurrence[{4,-5,1,2,0,-1,-1,2,-1}, {1,4,11,26,55,109,208,389,722}, 80] (* Vincenzo Librandi, Aug 17 2017 *)

SageMath

def A290989_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^6*(1+x^3)/((1-x)*(1-x+x^4)*(1-2*x+x^3-x^4)) ).list()
a=A290989_list(50); a[6:] # G. C. Greubel, Apr 12 2023

Formula

G.f.: x^6*(1 + x)*(1 - x + x^2)/((1 - x)*(1 - 2*x + x^3 - x^4)*(1 - x + x^4)).

a(n) = -2 + (1/19)*( 9*A099530(n+1) + 15*A099530(n) + 2*A099530(n-1) - A099530(n- 2) + 10*A059633(n+4) - 6*A059633(n+3) - 16*A059633(n+2) - A059633(n+1) ). - G. C. Greubel, Apr 12 2023