A027772 - OEIS

%I #30 Jan 30 2022 04:22:36

%S 12,169,1274,6825,29120,105196,334152,957372,2519400,6172530,14226212,

%T 31097794,64899744,130007500,251100200,469364220,851809140,1504982115,

%U 2594796750,4374736275,7225370880,11708971560,18644037360,29205813000,45060397200,68541870852

%N a(n) = (n+1)*binomial(n+1,12).

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

%H T. D. Noe, <a href="/A027772/b027772.txt">Table of n, a(n) for n = 11..1000</a>

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

%F G.f.: (12+x)*x^11/(1-x)^14.

%F From _Amiram Eldar_, Jan 30 2022: (Start)

%F Sum_{n>=11} 1/a(n) = 634871227/32016600 - 2*Pi^2.

%F Sum_{n>=11} (-1)^(n+1)/a(n) = Pi^2 + 5869568*log(2)/1155 - 113091604693/32016600. (End)

%t Table[(n+1)Binomial[n+1,12],{n,11,40}] (* or *) LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1},{12,169,1274,6825,29120,105196,334152,957372,2519400,6172530,14226212,31097794,64899744,130007500},30] (* _Harvey P. Dale_, Mar 13 2018 *)

%K nonn,easy

%O 11,1

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