A056117 Expansion of (1+8*x)/(1-x)^9.

1, 17, 117, 525, 1815, 5247, 13299, 30459, 64350, 127270, 238238, 425646, 730626, 1211250, 1947690, 3048474, 4657983, 6965343, 10214875, 14718275, 20868705, 29156985, 40190085, 54712125, 73628100, 98030556, 129229452, 168785452

Offset: 0

Author

Barry E. Williams, Jul 04 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A093644 ((9, 1) Pascal, column m=8). Partial sums of A052206.

Programs

GAP

List([0..30], n-> (9*n+8)*Binomial(n+7, 7)/8 ); # G. C. Greubel, Jan 18 2020

Magma

[(9*n+8)*Binomial(n+7, 7)/8: n in [0..30]]; // G. C. Greubel, Jan 18 2020

Maple

seq( (9*n+8)*binomial(n+7, 7)/8, n=0..30); # G. C. Greubel, Jan 18 2020

Mathematica

Table[9*Binomial[n+8,8] -8*Binomial[n+7,7], {n,0,30}] (* G. C. Greubel, Jan 18 2020 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,117,525,1815,5247,13299,30459,64350},30] (* Harvey P. Dale, Nov 23 2022 *)

PARI

vector(31, n, (9*n-1)*binomial(n+6, 7)/8) \\ G. C. Greubel, Jan 18 2020

Sage

[(9*n+8)*binomial(n+7, 7)/8 for n in (0..30)] # G. C. Greubel, Jan 18 2020

Formula

a(n) = (9*n+8)*binomial(n+7, 7)/8.

G.f.: (1+8*x)/(1-x)^9.

From G. C. Greubel, Jan 18 2020: (Start)

a(n) = 9*binomial(n+8,8) - 8*binomial(n+7,7).

E.g.f.: (40320 + 645120*x + 1693440*x^2 + 1505280*x^3 + 588000*x^4 + 112896*x^5 + 10976*x^6 + 512*x^7 + 9*x^8)*exp(x)/40320. (End)