A089274 Fifth column of the Legendre-Stirling triangle A071951.

1, 70, 3192, 121424, 4203824, 137922336, 4380918784, 136378114048, 4191383868672, 127754693361152, 3873052857829376, 117001609550671872, 3526270158211870720, 106112798944292282368, 3189880933574260359168

Offset: 0

Author

Wolfdieter Lang, Nov 07 2003

Keywords

• nonn
• easy

Comments

This is the fifth member of the family A000079 (powers of 2), A016129, A016309, A071952, etc.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (70,-1708,17544,-72000,86400).

Crossrefs

Cf. A016129, A016309, A071951, A071952, A089278, A089500.

Cf. A000079 (powers of 2).

Programs

Magma

[(16875*(6*5)^n - 20000*(5*4)^n + 6048*(4*3)^n - 405*(3*2)^n + 2*(2*1)^n)/2520: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011

Mathematica

Table[2^(n-3)*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)/315, {n,0,30}] (* G. C. Greubel, Nov 10 2024 *)

SageMath

def A089274(n): return 2^n*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)//2520
[A089274(n) for n in range(31)] # G. C. Greubel, Nov 10 2024

Formula

G.f.: 1/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)*(1-6*5*x)).

a(n) = (16875*(6*5)^n -20000*(5*4)^n +6048*(4*3)^n -405*(3*2)^n +2*(2*1)^n)/2520.

a(n) = A071951(n+5, 5), n>=0.

a(n) = det(|ps(i+5,j+4)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). [Mircea Merca, Apr 06 2013]

E.g.f.: (1/2520)*(2*exp(2*x) - 405*exp(6*x) + 6048*exp(12*x) - 20000*exp(20*x) + 16875*exp(30*x)). - G. C. Greubel, Nov 10 2024