A027807 - OEIS

%I #23 Feb 03 2022 05:04:14

%S 330,4455,32175,165165,675675,2342340,7147140,19691100,49884120,

%T 117781950,261891630,552882330,1115464350,2162284740,4045090500,

%U 7330194300,12907516050,22145248125,37105593525,60841155375,97796523825,154345534200,239501691000,365847510600

%N a(n) = 165*(n+1)*binomial(n+4,11)/4.

%C Number of 16-subsequences of [ 1, n ] with just 4 contiguous pairs.

%H T. D. Noe, <a href="/A027807/b027807.txt">Table of n, a(n) for n = 7..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

%F G.f.: 165*(2+x)*x^7/(1-x)^13.

%F a(n) = C(n+1, 8)*C(n+4, 4). - _Zerinvary Lajos_, May 26 2005; corrected by _R. J. Mathar_, Mar 16 2016

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

%F Sum_{n>=7} 1/a(n) = 10446643/198450 - 16*Pi^2/3.

%F Sum_{n>=7} (-1)^(n+1)/a(n) = 8*Pi^2/3 + 8192*log(2)/63 - 23108957/198450. (End)

%t Table[165 (n + 1) Binomial[n + 4, 11]/4, {n, 7, 30}] (* or *) Table[Binomial[n + 1, 8] Binomial[n + 4, 4], {n, 7, 30}] (* _Michael De Vlieger_, Mar 16 2016 *)

%K nonn,easy

%O 7,1

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