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Expansion of (1+3*x)/(1-x)^10.
3

%I #19 Feb 17 2023 10:04:22

%S 1,13,85,385,1375,4147,11011,26455,58630,121550,238238,445094,797810,

%T 1379210,2309450,3759074,5965487,9253475,14060475,20967375,30735705,

%U 44352165,63081525,88529025,122713500,168152556,227961228,305965660,406833460,536222500,700950052

%N Expansion of (1+3*x)/(1-x)^10.

%C Partial sums of A052181.

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H G. C. Greubel, <a href="/A055843/b055843.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = (4*n+9)*binomial(n+8, 8)/9.

%F G.f.: (1+3*x)/(1-x)^10.

%F a(n) = 4*binomial(n+9,9) - 3*binomial(n+8,8). - _G. C. Greubel_, Jan 21 2020

%F Sum_{n>=0} 1/a(n) = 9437184*Pi/24035 + 56623104*log(2)/24035 - 482087736/168245. - _Amiram Eldar_, Feb 17 2023

%p seq( (4*n+9)*binomial(n+8, 8)/9, n=0..30); # _G. C. Greubel_, Jan 21 2020

%t Table[4*Binomial[n+9,9] - 3*Binomial[n+8,8], {n,0,30}] (* _G. C. Greubel_, Jan 21 2020 *)

%o (PARI) vector(31, n, (4*n+5)*binomial(n+7, 8)/9) \\ _G. C. Greubel_, Jan 21 2020

%o (Magma) [(4*n+9)*Binomial(n+8, 8)/9: n in [0..30]]; // _G. C. Greubel_, Jan 21 2020

%o (Sage) [(4*n+9)*binomial(n+8, 8)/9 for n in (0..30)] # _G. C. Greubel_, Jan 21 2020

%o (GAP) List([0..30], n-> (4*n+9)*Binomial(n+8, 8)/9 ); # _G. C. Greubel_, Jan 21 2020

%Y Cf. A052181.

%Y Cf. A093561 ((4, 1) Pascal, column m=9).

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, May 30 2000