A139798 Coefficient of x^5 in (1-x-x^2)^(-n).

8, 38, 111, 256, 511, 924, 1554, 2472, 3762, 5522, 7865, 10920, 14833, 19768, 25908, 33456, 42636, 53694, 66899, 82544, 100947, 122452, 147430, 176280, 209430, 247338, 290493, 339416, 394661, 456816, 526504, 604384, 691152, 787542

Offset: 1

Author

Sergio Falcon, May 22 2008

Keywords

• nonn
• easy

Comments

The coefficient of x^5 in (1-x-x^2)^(-n) is the coefficient of x^5 in (1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5)^n. Using the multinomial theorem one then finds that a(n) = n(n+1)(n+2)(n^2 + 27n + 132)/5!

The inverse binomial transform yields 8,30,43,29,9,1,0,0,... (0 continued) - R. J. Mathar, May 23 2008

References

• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000096, A006503, A006504.

Programs

Mathematica

a[n_] := n(n + 1)(n + 2)(n^2 + 27n + 132)/5! Do[Print[n, " ", a[n]], {n, 1, 25}]
LinearRecurrence[{6,-15,20,-15,6,-1},{8,38,111,256,511,924},40] (* Harvey P. Dale, Oct 13 2015 *)

PARI

a(n)=binomial(n+2,3)*(n^2+27*n+132)/20 \\ Charles R Greathouse IV, Jul 29 2011

Formula

a(n) = n(n+1)(n+2)(n^2 + 27n + 132)/5!

O.g.f.: x(3x-4)(x-2)/(1-x)^6. - R. J. Mathar, May 23 2008

Extensions

More terms from R. J. Mathar, May 23 2008