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A027814 a(n) = 126*(n+1)*binomial(n+5,9)/5.

%I A027814 #27 Mar 05 2025 03:01:41
%S A027814 126,1512,9702,44352,162162,504504,1387386,3459456,7963956,17153136,
%T A027814 34918884,67721472,125919612,225629712,391270572,658982016,1081142370,
%U A027814 1732250520,2716483770,4177293120,6309453150,9374044680,13716915030,19791233280,28184836680
%N A027814 a(n) = 126*(n+1)*binomial(n+5,9)/5.
%C A027814 Number of 15-subsequences of [ 1, n ] with just 5 contiguous pairs.
%H A027814 T. D. Noe, <a href="/A027814/b027814.txt">Table of n, a(n) for n = 4..1000</a>
%H A027814 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A027814 G.f.: 126*(1+x)*x^4/(1-x)^11.
%F A027814 a(n) = C(n+1, 5)*C(n+5, 5). - _Zerinvary Lajos_, Apr 18 2005; corrected by _R. J. Mathar_, Feb 10 2016
%F A027814 From _Amiram Eldar_, Feb 03 2022: (Start)
%F A027814 Sum_{n>=4} 1/a(n) = 25*Pi^2/6 - 1160419/28224.
%F A027814 Sum_{n>=4} (-1)^n/a(n) = 25*Pi^2/12 - 27625/1344. (End)
%t A027814 Table[126(n+1)Binomial[n+5,9]/5,{n,4,40}]  (* _Harvey P. Dale_, Mar 13 2011 *)
%o A027814 (Magma)
%o A027814 A027814:= func< n | 126*(n+1)*Binomial(n+5,9)/5 >;
%o A027814 [A027814(n): n in [4..50]]; // _G. C. Greubel_, Mar 05 2025
%o A027814 (SageMath)
%o A027814 def A027814(n): return 126*(n+1)*binomial(n+5,9)//5
%o A027814 print([A027814(n) for n in range(4,41)]) # _G. C. Greubel_, Mar 05 2025
%Y A027814 Cf. A062264.
%K A027814 nonn,easy
%O A027814 4,1
%A A027814 Thi Ngoc Dinh (via _R. K. Guy_)