A395908 - OEIS

%I #22 May 16 2026 04:01:19

%S 0,27,175,660,1890,4550,9702,18900,34320,58905,96525,152152,232050,

%T 343980,497420,703800,976752,1332375,1789515,2370060,3099250,4006002,

%U 5123250,6488300,8143200,10135125,12516777,15346800,18690210,22618840,27211800,32555952,38746400,45886995,54090855,63480900,74190402

%N Series expansion of (27*x - 14*x^2 + 2*x^3) / (1 - x)^7.

%H Vincenzo Librandi, <a href="/A395908/b395908.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = A354794(n + 3, n).

%F a(n) = n * (n + 1) * (n + 2) * (n + 3) * (n + 5) * (n + 8) / 48.

%F From _Amiram Eldar_, May 15 2026: (Start)

%F Sum_{n>=1} 1/a(n) = 3991/88200.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 64*log(2)/21 - 36691/17640. (End)

%p gf := x*(2*x^2 - 14*x + 27)/(1 - x)^7: ser := series(gf, x, 33):

%p seq(coeff(ser, x, n), n = 0..32);

%t CoefficientList[Series[(2*x^3-14*x^2+27*x)/(1-x)^7,{x,0,40}],x] (* _Vincenzo Librandi_, May 11 2026 *)

%o (Python)

%o def a(n: int) -> int:

%o     return n * (n + 1) * (n + 2) * (n + 3) * (n + 5) * (n + 8) // 48

%o L = [a(n) for n in range(33)]; print(L)

%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((27*x - 14*x^2 + 2*x^3) / (1 - x)^7)); // _Vincenzo Librandi_, May 11 2026

%Y Cf. A354794, A395929.

%K nonn,easy

%O 0,2

%A _Peter Luschny_, May 10 2026