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A second-order recursive sequence.
4

%I #33 Jan 15 2023 02:54:19

%S 1,16,108,480,1650,4752,12012,27456,57915,114400,213928,381888,655044,

%T 1085280,1744200,2728704,4167669,6229872,9133300,13156000,18648630,

%U 26048880,35897940,48859200,65739375,87512256,115345296,150629248,195011080,250430400,319159632

%N A second-order recursive sequence.

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

%H Reinhard Zumkeller, <a href="/A056001/b056001.txt">Table of n, a(n) for n = 0..10000</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) = (n+1)*C(n+7, 7).

%F G.f.: (1+7*x)/(1-x)^9.

%F a(n) = A245334(n+7,7)/A000142(7). - _Reinhard Zumkeller_, Aug 31 2014

%F a(n) = A000581(n+8)+7*A000581(n+7). - _R. J. Mathar_, Oct 24 2014

%F E.g.f.: (5040 +75600*x +194040*x^2 +170520*x^3 +66150*x^4 +12642*x^5 + 1225*x^6 +57*x^7 +x^8)*exp(x)/5040. - _G. C. Greubel_, Aug 29 2019

%F From _Amiram Eldar_, Jan 15 2023: (Start)

%F Sum_{n>=0} 1/a(n) = 7*Pi^2/6 - 37583/3600.

%F Sum_{n>=0} (-1)^n/a(n) = 7*Pi^2/12 - 2912*log(2)/15 + 155701/1200. (End)

%p seq((n+1)*binomial(n+7,7), n=0..30); # _G. C. Greubel_, Aug 29 2019

%t Table[(n+1)Binomial[n+7, 7], {n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 19 2011; corrected by _Bruno Berselli_, Jan 23 2015 *)

%o (Haskell)

%o a056001 n = (n + 1) * a007318' (n + 7) 7

%o -- _Reinhard Zumkeller_, Aug 31 2014

%o (PARI) vector(30, n, n*binomial(n+6,7)) \\ _G. C. Greubel_, Aug 29 2019

%o (Magma) [(n+1)*Binomial(n+7,7): n in [0..30]]; // _G. C. Greubel_, Aug 29 2019

%o (Sage) [(n+1)*binomial(n+7,7) for n in (0..30)] # _G. C. Greubel_, Aug 29 2019

%o (GAP) List([0..30], n-> (n+1)*Binomial(n+7,7)); # _G. C. Greubel_, Aug 29 2019

%Y Partial sums of A052226.

%Y Cf. A093565 ((8, 1) Pascal, column m=8).

%Y Cf. A007318, A000142, A245334.

%K nonn,easy

%O 0,2

%A _Barry E. Williams_, Jun 18 2000