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A293475 a(n) = (3*n + 4)*Pochhammer(n, 4) / 4!.

%I A293475 #22 Aug 31 2025 02:53:13
%S A293475 0,7,50,195,560,1330,2772,5250,9240,15345,24310,37037,54600,78260,
%T A293475 109480,149940,201552,266475,347130,446215,566720,711942,885500,
%U A293475 1091350,1333800,1617525,1947582,2329425,2768920,3272360,3846480,4498472,5236000,6067215,7000770
%N A293475 a(n) = (3*n + 4)*Pochhammer(n, 4) / 4!.
%H A293475 G. C. Greubel, <a href="/A293475/b293475.txt">Table of n, a(n) for n = 0..1000</a>
%H A293475 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A293475 a(n) = n*Stirling2(3 + n, 1 + n).
%F A293475 -a(-n-3) = (n + 3)*abs(Stirling1(n+2, n)) for n >= 0.
%F A293475 -a(-n-3) = a(n) + binomial(n+3, 4) for n >= 0.
%F A293475 From _Colin Barker_, Nov 21 2017: (Start)
%F A293475 G.f.: x*(7 + 8*x) / (1 - x)^6.
%F A293475 a(n) = n*(24 + 62*n + 57*n^2 + 22*n^3 + 3*n^4)/24.
%F A293475 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 A293475 From _Amiram Eldar_, Aug 31 2025: (Start)
%F A293475 Sum_{n>=1} 1/a(n) = 27*sqrt(3)*Pi/10 + 243*log(3)/10 - 2473/60.
%F A293475 Sum_{n>=1} (-1)^(n+1)/a(n) = 27*sqrt(3)*Pi/5 + 112*log(2)/5 - 2687/60. (End)
%p A293475 A293475 := n -> (3*n + 4)*pochhammer(n, 4)/4!:
%p A293475 seq(A293475(n), n=0..32);
%t A293475 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 7, 50, 195, 560, 1330}, 32]
%t A293475 Table[n*StirlingS2[n+3, n+1], {n, 0, 50}] (* _G. C. Greubel_, Nov 20 2017 *)
%t A293475 f[n_] := (3n + 4) Pochhammer[n, 4]/4!; Array[f, 35, 0] (* or *)
%t A293475 CoefficientList[ Series[ x (7 + 8x)/(1 - x)^6, {x, 0, 34}], x] (* _Robert G. Wilson v_, Nov 21 2017 *)
%o A293475 (PARI) for(n=0,30, print1(n*Stirling(n+3, n+1, 2), ", ")) \\ _G. C. Greubel_, Nov 20 2017
%o A293475 (Magma) [0] cat [(3*n + 4)*Factorial(n+3)/(Factorial(n-1)*Factorial(4)): n in [1..30]]; // _G. C. Greubel_, Nov 20 2017
%o A293475 (PARI) concat(0, Vec(x*(7 + 8*x) / (1 - x)^6 + O(x^40))) \\ _Colin Barker_, Nov 21 2017
%Y A293475 Cf. A265609, A293476, A293608, A293615.
%K A293475 nonn,easy,changed
%O A293475 0,2
%A A293475 _Peter Luschny_, Oct 20 2017