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A027818 a(n) = (n+1)*binomial(n+6,6).

%I A027818 #39 Jan 28 2022 04:30:10
%S A027818 1,14,84,336,1050,2772,6468,13728,27027,50050,88088,148512,241332,
%T A027818 379848,581400,868224,1268421,1817046,2557324,3542000,4834830,6512220,
%U A027818 8665020,11400480,14844375,19143306,24467184,31011904,39002216,48694800,60381552,74393088,91102473
%N A027818 a(n) = (n+1)*binomial(n+6,6).
%C A027818 Number of 13-subsequences of [ 1, n ] with just 6 contiguous pairs.
%H A027818 Reinhard Zumkeller, <a href="/A027818/b027818.txt">Table of n, a(n) for n = 0..10000</a>
%H A027818 <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 A027818 G.f.: (1+6*x)/(1-x)^8.
%F A027818 a(n) = A245334(n+6,6)/A000142(6). - _Reinhard Zumkeller_, Aug 31 2014
%F A027818 E.g.f.: (7! +9360*x +20520*x^2 +15000*x^3 +4650*x^4 +666*x^5 +43*x^6 + x^7)*exp(x)/7!. - _G. C. Greubel_, Aug 29 2019
%F A027818 From _Amiram Eldar_, Jan 28 2022: (Start)
%F A027818 Sum_{n>=0} 1/a(n) = Pi^2 - 5269/600.
%F A027818 Sum_{n>=0} (-1)^n/a(n) = Pi^2/2 - 512*log(2)/5 + 40189/600. (End)
%p A027818 seq((n+1)*binomial(n+6,6), n=0..30); # _Zerinvary Lajos_, Oct 19 2006
%t A027818 Table[(n+1)*Binomial[n+6,6], {n,0,30}] (* _G. C. Greubel_, Aug 29 2019 *)
%o A027818 (Haskell)
%o A027818 a027818 n = (n + 1) * a007318' (n + 6) 6
%o A027818 -- _Reinhard Zumkeller_, Aug 31 2014
%o A027818 (PARI) a(n) = (n+1)*binomial(n+6,6) \\ _Charles R Greathouse IV_, Jun 11 2015
%o A027818 (Magma) [(n+1)*Binomial(n+6,6): n in [0..30]]; // _G. C. Greubel_, Aug 29 2019
%o A027818 (Sage) [(n+1)*binomial(n+6,6) for n in (0..30)] # _G. C. Greubel_, Aug 29 2019
%o A027818 (GAP) List([0..30], n-> (n+1)*Binomial(n+6,6)); # _G. C. Greubel_, Aug 29 2019
%Y A027818 Cf. A093564 ((7, 1) Pascal, column m=7). Partial sums of A050403.
%Y A027818 Cf. A062190, A007318, A000142, A245334.
%K A027818 nonn,easy
%O A027818 0,2
%A A027818 Thi Ngoc Dinh (via _R. K. Guy_)