A191596 Expansion of (1+x)^4/(1-x)^7.

1, 11, 62, 242, 743, 1925, 4396, 9108, 17469, 31471, 53834, 88166, 139139, 212681, 316184, 458728, 651321, 907155, 1241878, 1673882, 2224607, 2918861, 3785156, 4856060, 6168565, 7764471, 9690786, 12000142, 14751227, 18009233, 21846320

Offset: 0

Author

Bruno Berselli, Jun 08 2011

Keywords

• nonn
• easy

Comments

The first, second and third differences are in A069038, A001846 and A008412, respectively.

Inverse binomial transform of this sequence: 1, 10, 41, 88, 104, 64, 16, 0, 0 (0 continued).

Also (by Superseeker), the n-th coefficient of the expansion of ((1+x)^4/(1-x)^7)*(1+x)^n is A006976(n-1).

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013
• M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A008415, A001848, A069039, A008412, A001846, A069038, A061927 (for type of g.f.).

Programs

Magma

[(2*n^6+18*n^5+80*n^4+210*n^3+323*n^2+267*n+90)/90: n in [0..30]]; // Vincenzo Librandi, Jun 08 2011

Maple

A191596:=n->(n+1)*(n+2)*(2*n^4+12*n^3+40*n^2+66*n+45)/90: seq(A191596(n), n=0..40); # Wesley Ivan Hurt, Nov 20 2014

Mathematica

CoefficientList[Series[(1 + x)^4/(1 - x)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Nov 20 2014 *)

Maxima

makelist(coeff(taylor((1+x)^4/(1-x)^7, x, 0, n), x, n), n, 0, 30);

PARI

a(n)=(((((n+n+18)*n+80)*n+210)*n+323)*n+267)/90*n+1 \\ Charles R Greathouse IV, Jun 08 2011

Formula

G.f.: (1+x)^4/(1-x)^7.

a(n) = (n+1)*(n+2)*(2*n^4+12*n^3+40*n^2+66*n+45)/90.

a(n) = a(-n-3) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).

By Superseeker:

a(n)+a(n+1) = A069039(n+2),

a(n+2)-a(n) = A001847(n+2),

a(n+2)+2*a(n+1)+a(n) = A001848(n+2).