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

0,2

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).

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)

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 *)

Prog

(PARI) vector(31, n, (9*n-1)*binomial(n+6, 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
(SageMath) [(9*n+8)*binomial(n+7, 7)/8 for n in (0..30)] # G. C. Greubel, Jan 18 2020
(GAP) List([0..30], n-> (9*n+8)*Binomial(n+7, 7)/8 ); # G. C. Greubel, Jan 18 2020

Crossrefs

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

Sequence in context: A044349 A044730 A086020 * A196575 A003109 A066607

Adjacent sequences: A056114 A056115 A056116 * A056118 A056119 A056120

Keyword

easy,nonn

Author

Barry E. Williams, Jul 04 2000

Status

approved