A395908 Series expansion of (27*x - 14*x^2 + 2*x^3) / (1 - x)^7.

0, 27, 175, 660, 1890, 4550, 9702, 18900, 34320, 58905, 96525, 152152, 232050, 343980, 497420, 703800, 976752, 1332375, 1789515, 2370060, 3099250, 4006002, 5123250, 6488300, 8143200, 10135125, 12516777, 15346800, 18690210, 22618840, 27211800, 32555952, 38746400, 45886995, 54090855, 63480900, 74190402

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

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

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

From Amiram Eldar, May 15 2026: (Start)

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

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

Maple

gf := x*(2*x^2 - 14*x + 27)/(1 - x)^7: ser := series(gf, x, 33):
seq(coeff(ser, x, n), n = 0..32);

Mathematica

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

Prog

(Python)
def a(n: int) -> int:
    return n * (n + 1) * (n + 2) * (n + 3) * (n + 5) * (n + 8) // 48
L = [a(n) for n in range(33)]; print(L)
(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

Crossrefs

Cf. A354794, A395929.

Sequence in context: A174617 A055339 A269054 * A248234 A083560 A125337

Adjacent sequences: A395905 A395906 A395907 * A395909 A395917 A395918

Keyword

nonn,easy

Author

Peter Luschny, May 10 2026

Status

approved