A293476 a(n) = ((n + 1)/2)*(n + 2)*Pochhammer(n, 5) / 4!.
0, 15, 180, 1050, 4200, 13230, 35280, 83160, 178200, 353925, 660660, 1171170, 1987440, 3248700, 5140800, 7907040, 11860560, 17398395, 25017300, 35331450, 49092120, 67209450, 90776400, 121095000, 159705000, 208415025, 269336340, 344919330, 437992800, 551806200
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Programs
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Magma
[0] cat [((n + 1)/2)*(n + 2)*Factorial(n+4)/(Factorial(4)*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017
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Maple
A293476 := n -> ((n+1)/2)*(n+2)*pochhammer(n, 5)/4!: seq(A293476(n), n=0..11);
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Mathematica
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 15, 180, 1050, 4200, 13230, 35280, 83160}, 32] Table[n*StirlingS2[4 + n, 1 + n], {n,0,50}] (* G. C. Greubel, Nov 20 2017 *) -
PARI
for(n=0, 30, print1(n*stirling(n+4, n+1, 2), ", ")) \\ G. C. Greubel, Nov 20 2017
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PARI
concat(0, Vec(15*x*(1 + 4*x + 2*x^2) / (1 - x)^8 + O(x^40))) \\ Colin Barker, Nov 21 2017
Formula
a(n) = n*Stirling2(4 + n, 1 + n).
-a(-n-4) = (n+4)*abs(Stirling1(n+3, n)) for n >= 0.
-a(-n-4) = a(n) + 5*binomial(n+4, 5)*(n+2) for n >= 0.
From Colin Barker, Nov 21 2017: (Start)
G.f.: 15*x*(1 + 4*x + 2*x^2) / (1 - x)^8.
a(n) = (1/48)*(n*(2 + 3*n + n^2)^2*(12 + 7*n + n^2)).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 1187/36 - 10*Pi^2/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 653/36 - Pi^2/3 - 64*log(2)/3. (End)