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

0, 50, 390, 1680, 5320, 13860, 31500, 64680, 122760, 218790, 370370, 600600, 939120, 1423240, 2099160, 3023280, 4263600, 5901210, 8031870, 10767680, 14238840, 18595500, 24009700, 30677400, 38820600, 48689550, 60565050, 74760840, 91626080, 111547920, 134954160

Offset: 0

Author

Peter Luschny, Oct 20 2017

Keywords

• nonn
• easy
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A265609, A293475, A293476, A293615.

Programs

Magma

[0] cat [(3*n + 7)*Factorial(n+4)/(Factorial(4)*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017

Maple

A293608 := n -> (3*n+7)*pochhammer(n, 5)/4!:
seq(A293608(n), n=0..11);

Mathematica

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 50, 390, 1680, 5320, 13860, 31500}, 32]
Table[n*(n+1)*StirlingS2[4 + n, 2 + n], {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)
f[n_] := (3 n + 7) Pochhammer[n, 5]/4!; Array[f, 31, 0] (* or *)
CoefficientList[ Series[10x (5 + 4x)/(1 - x)^7, {x, 0, 30}], x] (* Robert G. Wilson v, Nov 21 2017 *)

PARI

for(n=0,30, print1(n*(n+1)*stirling(4 + n, 2 + n, 2), ", ")) \\ G. C. Greubel, Nov 20 2017

PARI

concat(0, Vec(10*x*(5 + 4*x) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Nov 21 2017

Formula

a(n) = n*(n+1)*Stirling2(4 + n, 2 + n).

-a(-n-4) = a(n) - 30*binomial(n+4, 5)*(n + 2) for n >= 0.

From Colin Barker, Nov 21 2017: (Start)

G.f.: 10*x*(5 + 4*x) / (1 - x)^7.

a(n) = (1/24)*(n*(168 + 422*n + 395*n^2 + 175*n^3 + 37*n^4 + 3*n^5)).

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)

From Amiram Eldar, Aug 31 2025: (Start)

Sum_{n>=1} 1/a(n) = 8703/490 - 81*sqrt(3)*Pi/70 - 729*log(3)/70.

Sum_{n>=1} (-1)^(n+1)/a(n) = 81*sqrt(3)*Pi/35 + 416*log(2)/35 - 15298/735. (End)