%I #18 Sep 08 2022 08:46:19

%S 0,7,50,195,560,1330,2772,5250,9240,15345,24310,37037,54600,78260,

%T 109480,149940,201552,266475,347130,446215,566720,711942,885500,

%U 1091350,1333800,1617525,1947582,2329425,2768920,3272360,3846480,4498472,5236000,6067215,7000770

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

%H G. C. Greubel, <a href="/A293475/b293475.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = n*Stirling2(3 + n, 1 + n).

%F -a(-n-3) = (n + 3)*abs(Stirling1(n+2, n)) for n >= 0.

%F -a(-n-3) = a(n) + binomial(n+3, 4) for n >= 0.

%F From _Colin Barker_, Nov 21 2017: (Start)

%F G.f.: x*(7 + 8*x) / (1 - x)^6.

%F a(n) = n*(24 + 62*n + 57*n^2 + 22*n^3 + 3*n^4)/24.

%F 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)

%p A293475 := n -> (3*n + 4)*pochhammer(n, 4)/4!:

%p seq(A293475(n), n=0..32);

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 7, 50, 195, 560, 1330}, 32]

%t Table[n*StirlingS2[n+3, n+1], {n, 0, 50}] (* _G. C. Greubel_, Nov 20 2017 *)

%t f[n_] := (3n + 4) Pochhammer[n, 4]/4!; Array[f, 35, 0] (* or *)

%t CoefficientList[ Series[ x (7 + 8x)/(1 - x)^6, {x, 0, 34}], x] (* _Robert G. Wilson v_, Nov 21 2017 *)

%o (PARI) for(n=0,30, print1(n*Stirling(n+3, n+1, 2), ", ")) \\ _G. C. Greubel_, Nov 20 2017

%o (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 (PARI) concat(0, Vec(x*(7 + 8*x) / (1 - x)^6 + O(x^40))) \\ _Colin Barker_, Nov 21 2017

%Y Cf. A265609, A293476, A293608, A293615.

%K nonn,easy

%O 0,2

%A _Peter Luschny_, Oct 20 2017