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%I #18 Nov 23 2022 14:55:44
%S 1,17,117,525,1815,5247,13299,30459,64350,127270,238238,425646,730626,
%T 1211250,1947690,3048474,4657983,6965343,10214875,14718275,20868705,
%U 29156985,40190085,54712125,73628100,98030556,129229452,168785452
%N Expansion of (1+8*x)/(1-x)^9.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H G. C. Greubel, <a href="/A056117/b056117.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = (9*n+8)*binomial(n+7, 7)/8.
%F G.f.: (1+8*x)/(1-x)^9.
%F From _G. C. Greubel_, Jan 18 2020: (Start)
%F a(n) = 9*binomial(n+8,8) - 8*binomial(n+7,7).
%F 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)
%p seq( (9*n+8)*binomial(n+7, 7)/8, n=0..30); # _G. C. Greubel_, Jan 18 2020
%t Table[9*Binomial[n+8,8] -8*Binomial[n+7,7], {n,0,30}] (* _G. C. Greubel_, Jan 18 2020 *)
%t 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 *)
%o (PARI) vector(31, n, (9*n-1)*binomial(n+6, 7)/8) \\ _G. C. Greubel_, Jan 18 2020
%o (Magma) [(9*n+8)*Binomial(n+7, 7)/8: n in [0..30]]; // _G. C. Greubel_, Jan 18 2020
%o (Sage) [(9*n+8)*binomial(n+7, 7)/8 for n in (0..30)] # _G. C. Greubel_, Jan 18 2020
%o (GAP) List([0..30], n-> (9*n+8)*Binomial(n+7, 7)/8 ); # _G. C. Greubel_, Jan 18 2020
%Y Cf. A093644 ((9, 1) Pascal, column m=8). Partial sums of A052206.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jul 04 2000
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