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A293608 a(n) = (3*n + 7)*Pochhammer(n, 5) / 4!.

%I A293608 #22 Aug 31 2025 02:49:19
%S A293608 0,50,390,1680,5320,13860,31500,64680,122760,218790,370370,600600,
%T A293608 939120,1423240,2099160,3023280,4263600,5901210,8031870,10767680,
%U A293608 14238840,18595500,24009700,30677400,38820600,48689550,60565050,74760840,91626080,111547920,134954160
%N A293608 a(n) = (3*n + 7)*Pochhammer(n, 5) / 4!.
%H A293608 G. C. Greubel, <a href="/A293608/b293608.txt">Table of n, a(n) for n = 0..1000</a>
%H A293608 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A293608 a(n) = n*(n+1)*Stirling2(4 + n, 2 + n).
%F A293608 -a(-n-4) = a(n) - 30*binomial(n+4, 5)*(n + 2) for n >= 0.
%F A293608 From _Colin Barker_, Nov 21 2017: (Start)
%F A293608 G.f.: 10*x*(5 + 4*x) / (1 - x)^7.
%F A293608 a(n) = (1/24)*(n*(168 + 422*n + 395*n^2 + 175*n^3 + 37*n^4 + 3*n^5)).
%F A293608 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6. (End)
%F A293608 From _Amiram Eldar_, Aug 31 2025: (Start)
%F A293608 Sum_{n>=1} 1/a(n) = 8703/490 - 81*sqrt(3)*Pi/70 - 729*log(3)/70.
%F A293608 Sum_{n>=1} (-1)^(n+1)/a(n) = 81*sqrt(3)*Pi/35 + 416*log(2)/35 - 15298/735. (End)
%p A293608 A293608 := n -> (3*n+7)*pochhammer(n, 5)/4!:
%p A293608 seq(A293608(n), n=0..11);
%t A293608 LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 50, 390, 1680, 5320, 13860, 31500}, 32]
%t A293608 Table[n*(n+1)*StirlingS2[4 + n, 2 + n], {n,0,50}] (* _G. C. Greubel_, Nov 20 2017 *)
%t A293608 f[n_] := (3 n + 7) Pochhammer[n, 5]/4!; Array[f, 31, 0] (* or *)
%t A293608 CoefficientList[ Series[10x (5 + 4x)/(1 - x)^7, {x, 0, 30}], x] (* _Robert G. Wilson v_, Nov 21 2017 *)
%o A293608 (PARI) for(n=0,30, print1(n*(n+1)*stirling(4 + n, 2 + n, 2), ", ")) \\ _G. C. Greubel_, Nov 20 2017
%o A293608 (Magma) [0] cat [(3*n + 7)*Factorial(n+4)/(Factorial(4)*Factorial(n-1)): n in [1..30]]; // _G. C. Greubel_, Nov 20 2017
%o A293608 (PARI) concat(0, Vec(10*x*(5 + 4*x) / (1 - x)^7 + O(x^40))) \\ _Colin Barker_, Nov 21 2017
%Y A293608 Cf. A265609, A293475, A293476, A293615.
%K A293608 nonn,easy,changed
%O A293608 0,2
%A A293608 _Peter Luschny_, Oct 20 2017