%I #24 Feb 04 2022 09:03:03
%S 15,126,588,2016,5670,13860,30492,61776,117117,210210,360360,594048,
%T 946764,1465128,2209320,3255840,4700619,6662502,9287124,12751200,
%U 17267250,23088780,30515940,39901680,51658425,66265290,84275856,106326528,133145496,165562320,204518160
%N a(n) = 3*(n+1)*binomial(n+2,6).
%C Number of 9-subsequences of [ 1, n ] with just 2 contiguous pairs.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F G.f.: 3*(5+2x)*x^4/(1-x)^8.
%F a(n) = C(n+1, 5)*C(n+2, 2). - _Zerinvary Lajos_, May 13 2005; corrected by _R. J. Mathar_, Feb 10 2016
%F From _Amiram Eldar_, Feb 04 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 5*Pi^2/3 - 2947/180.
%F Sum_{n>=4} (-1)^n/a(n) = 5*Pi^2/6 + 128*log(2)/3 - 6793/180. (End)
%p [seq (stirling2(n+1,n)*binomial(n,5),n=5..29)]; # _Zerinvary Lajos_, Dec 06 2006
%t Table[3 * (n+1) * Binomial[n+2, 6], {n, 4, 50}] (* _Amiram Eldar_, Feb 04 2022 *)
%K nonn,easy
%O 4,1
%A Thi Ngoc Dinh (via _R. K. Guy_)