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

0,2

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).

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

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 *)

Prog

(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
(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

Crossrefs

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

Cf. A000079 (powers of 2).

Sequence in context: A069296 A349466 A075923 * A266739 A227273 A298463

Adjacent sequences: A089271 A089272 A089273 * A089275 A089276 A089277

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 07 2003

Status

approved