A229702 Expansion of 1/((1-x)^4*(1-6x)).

1, 10, 70, 440, 2675, 16106, 96720, 580440, 3482805, 20897050, 125382586, 752295880, 4513775735, 27082654970, 162495930500, 974975583816, 5849853503865, 35099121024330, 210594726147310, 1263568356885400, 7581410141314171

Offset

0,2

Comments

This sequence was chosen to illustrate a way to match generating functions and closed-form solutions.

The general term associated with the generating function

1/((1-s*x)^4*(1-r*x)) with r>s>=1 is a(n) = [ r^(n+4) - s^(n+1)*(s^3 + s^2*(r-s)*binomial(n+4,1) + s*(r-s)^2*binomial(n+4,2)+(r-s)^3*binomial(n+4,3))]/(r-s)^4.

Links

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (10,-30,40,-25,6).

Formula

a(n) = (6^(n+4) - (1 + 5*C(n+4,1) + 25*C(n+4,2) + 125*C(n+4,3)))/625 = (6^(n+5) - (125*n^3 + 1200*n^2 + 3805*n + 4026))/3750.

From Enrique Navarrete, Nov 11 2025: (Start)

a(n) = 10*a(n-1) - 30*a(n-2) + 40*a(n-3) - 25*a(n-4) + 6*a(n-5), n >= 5.

E.g.f.: exp(x)*(6^5*exp(5*x) - 125*x^3 - 1575*x^2 - 5130*x - 4026)/3750. (End)

Example

a(3) = (6^8 - (125*3^3  + 1200*3^2 + 3805*3 + 4026))/3750 = 440.

Crossrefs

Cf. A002663, A097786, A097788, A097790.

Partial sums of A390418; second partial sums of A014829; third partial sums of A003464; fourth partial sums of A000400.

Sequence in context: A122892 A125347 A005465 * A337992 A397669 A257114

Adjacent sequences: A229699 A229700 A229701 * A229703 A229704 A229705

Keyword

nonn,easy

Author

Yahia Kahloune, Sep 27 2013

Status

approved