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A293615 a(n) = Pochhammer(n, 5) / 2.

%I A293615 #24 Aug 31 2025 02:48:34
%S A293615 0,60,360,1260,3360,7560,15120,27720,47520,77220,120120,180180,262080,
%T A293615 371280,514080,697680,930240,1220940,1580040,2018940,2550240,3187800,
%U A293615 3946800,4843800,5896800,7125300,8550360,10194660,12082560,14240160,16695360,19477920,22619520
%N A293615 a(n) = Pochhammer(n, 5) / 2.
%H A293615 G. C. Greubel, <a href="/A293615/b293615.txt">Table of n, a(n) for n = 0..1000</a>
%H A293615 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A293615 a(n) = n*(n+1)*(n+2)*Stirling2(4 + n, 3 + n).
%F A293615 -a(-n-4) = a(n) for n >= 0.
%F A293615 a(n) = 60*A000389(n+4). - _G. C. Greubel_, Nov 20 2017
%F A293615 From _Colin Barker_, Nov 21 2017: (Start)
%F A293615 G.f.: 60*x / (1 - x)^6.
%F A293615 a(n) = (1/2)*(n*(1 + n)*(2 + n)*(3 + n)*(4 + n)).
%F A293615 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. (End)
%F A293615 From _Amiram Eldar_, Aug 31 2025: (Start)
%F A293615 Sum_{n>=1} 1/a(n) = 1/48.
%F A293615 Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 131/144. (End)
%p A293615 A293615 := n -> pochhammer(n, 5)/2:
%p A293615 seq(A293615(n), n=0..11);
%t A293615 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 60, 360, 1260, 3360, 7560}, 32]
%t A293615 Table[Pochhammer[n, 5]/2, {n,0,50}] (* _G. C. Greubel_, Nov 20 2017 *)
%o A293615 (PARI) for(n=0,30, print1(n*(n+1)*(n+2)*stirling(4 + n, 3 + n, 2), ", ")) \\ _G. C. Greubel_, Nov 20 2017
%o A293615 (Magma) [0] cat [Factorial(n+4)/(2*Factorial(n-1)): n in [1..30]]; // _G. C. Greubel_, Nov 20 2017
%o A293615 (PARI) concat(0, Vec(60*x / (1 - x)^6 + O(x^40))) \\ _Colin Barker_, Nov 21 2017
%Y A293615 Cf. A000389, A265609, A293475, A293476, A293608.
%K A293615 nonn,easy,changed
%O A293615 0,2
%A A293615 _Peter Luschny_, Oct 20 2017