A092055 a(n) = binomial(2 + 2^n,3).

1, 4, 20, 120, 816, 5984, 45760, 357760, 2829056, 22500864, 179481600, 1433753600, 11461636096, 91659526144, 733141975040, 5864598896640, 46914643623936, 375308558925824, 3002434111406080, 24019335451770880, 192154133857304576, 1537230871833083904, 12297838178567454720

Offset: 0

Author

Henry Bottomley, Feb 19 2004

Keywords

• nonn
• easy

Comments

a(n) = Sum_{i=1...(2^n)} i*(i+1)/2, this sequence is thus similar to A016131 as it is a sum of triangular numbers on the interval <1,2^n>, A016131 is a sum of triangular numbers on the interval <1,2^n - 1>. - Ctibor O. Zizka, Mar 03 2009

a(n) is the number of unordered (not necessarily distinct) triples of subsets taken from the power set of {1,2,...,n}. Cf. A007582 (pairs of such subsets). - Geoffrey Critzer, Jul 10 2013

Examples

			a(5) = C(2+2^5,3) = C(34,3) = 5984.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Crossrefs

Cf. A000079, A007581, A092056, A016131.

Programs

Magma

[Binomial(2^n+2, 3): n in [0..30]]; // G. C. Greubel, Dec 27 2024

Maple

seq(binomial(2+2^n, 3), n=0..25); # Zerinvary Lajos, Feb 22 2008

Mathematica

nn=20;Table[Coefficient[Series[1/(1-x)^(2^n),{x,0,nn}],x^3],{n,0,nn}] (* Geoffrey Critzer, Jul 10 2013 *)
Binomial[2+2^Range[0,30], 3] (* G. C. Greubel, Dec 27 2024 *)

PARI

Vec((1-10*x+20*x^2)/((1-2*x)*(1-4*x)*(1-8*x)) + O(x^100)) \\ Colin Barker, Sep 13 2014

Python

def A092055(n): return binomial(pow(2,n)+2,3)
print([A092055(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

Formula

a(n) = (2^(3*n-1) +3*2^(2*n-1) + 2^n)/3 = A092056(3, n).

a(n) = A000079(n)*A007581(n).

a(n) = 2*a(n-1) + 4^(n-1) + 8^(n-1).

a(n) = [x^3] 1/(1-x)^(2^n). - Geoffrey Critzer, Jul 11 2013

a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3). - Colin Barker, Sep 13 2014

G.f.: (1-10*x+20*x^2) / ((1-2*x)*(1-4*x)*(1-8*x)). - Colin Barker, Sep 13 2014

E.g.f.: (1/6)*(2*exp(2*x) + 3*exp(4*x) + exp(8*x)). - G. C. Greubel, Dec 27 2024