A027778 a(n) = 5*(n+1)*binomial(n+2, 5)/2.

10, 75, 315, 980, 2520, 5670, 11550, 21780, 38610, 65065, 105105, 163800, 247520, 364140, 523260, 736440, 1017450, 1382535, 1850695, 2443980, 3187800, 4111250, 5247450, 6633900, 8312850, 10331685, 12743325, 15606640, 18986880, 22956120, 27593720, 32986800, 39230730

Offset: 3

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 8-subsequences of [ 1, n ] with just 2 contiguous pairs.

Links

• Michael De Vlieger, Table of n, a(n) for n = 3..10000
• Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Journal of Integer Sequences, Vol. 24 (2021), Article 21.8.6; ResearchGate link.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Not equal to 5*A005715(n+1)/2.

Programs

Mathematica

DeleteCases[CoefficientList[Series[5 x^3*(2 + x)/(1 - x)^7, {x, 0, 24}], x], 0] (* Michael De Vlieger, Jul 16 2021 *)

Formula

G.f.: 5*x^3*(2+x)/(1-x)^7.

a(n) = binomial(n+1, 4)*binomial(n+2, 2). - Zerinvary Lajos, Apr 28 2005, corrected by R. J. Mathar, Feb 13 2016

From Amiram Eldar, Feb 01 2022: (Start)

Sum_{n>=3} 1/a(n) = 239/18 - 4*Pi^2/3.

Sum_{n>=3} (-1)^(n+1)/a(n) = 2*Pi^2/3 + 64*log(2)/3 - 383/18. (End)