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a(n) = 7*(n+1)*binomial(n+6,7)/2.
1

%I #17 Feb 03 2022 04:53:22

%S 7,84,504,2100,6930,19404,48048,108108,225225,440440,816816,1447992,

%T 2469012,4069800,6511680,10147368,15444891,23015916,33649000,48348300,

%U 68378310,95315220,131105520,178132500,239291325,318073392,418660704,546031024,706074600,905723280

%N a(n) = 7*(n+1)*binomial(n+6,7)/2.

%C Number of 14-subsequences of [ 1, n ] with just 6 contiguous pairs.

%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) = 7*A052181(n).

%F G.f.: 7*(1+3*x)*x/(1-x)^9.

%F a(n) = C(n+1,2)*C(n+6,6). - _Zerinvary Lajos_, May 26 2005

%F From _Amiram Eldar_, Feb 03 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 5969/300 - 2*Pi^2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 - 384*log(2)/5 + 13049/300. (End)

%t Table[7*(n + 1)*Binomial[n + 6, 7]/2, {n, 1, 50}] (* _Amiram Eldar_, Feb 03 2022 *)

%Y Cf. A052181.

%K nonn,easy

%O 1,1

%A Thi Ngoc Dinh (via _R. K. Guy_)