A038846 4-fold convolution of A000302 (powers of 4); expansion of g.f. 1/(1-4*x)^4.

1, 16, 160, 1280, 8960, 57344, 344064, 1966080, 10813440, 57671680, 299892736, 1526726656, 7633633280, 37580963840, 182536110080, 876173328384, 4161823309824, 19585050869760, 91396904058880, 423311976693760, 1947235092791296, 8901646138474496, 40462027902156800

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also minimal 3-covers of a labeled n-set that cover 3 points of that set uniquely (if offset is 3). Cf. A057524 for unlabeled case. - Vladeta Jovovic, Sep 02 2000

Also convolution of A020918 with A000984 (central binomial coefficients).

Let M=[1,0,0,i;0,1,i,0;0,i,1,0;i,0,0,1], i=sqrt(-1). Then 1/det(I-xM) = 1/(1-4x)^4. - Paul Barry, Apr 27 2005

With a different offset, number of n-permutations (n=4) of 5 objects u, v, w, z, x with repetition allowed, containing exactly three u's. Example: a(1)=16 because we have uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu and xuuu. - Zerinvary Lajos, May 19 2008

From A152818. a(n) = A006044/6. - Paul Curtz, Jan 07 2009

Also convolution of A000302 with A038845, also convolution of A002457 with A002802, also convolution of A002697. - Rui Duarte, Oct 08 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Crossrefs

Cf. A000302, A000984, A002457, A002697, A002802, A006044, A020918, A038231, A038845, A057524, A152818.

Programs

GAP

List([0..30], n-> 4^n*Binomial(n+3,3) ) # G. C. Greubel, Jul 20 2019

Magma

[4^n*Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(seq(binomial(i, j)*4^(i-3), j =i-3), i=3..33); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+3,3)*4^n,n=0..30); # Zerinvary Lajos, May 19 2008

Mathematica

Table[4^n*Binomial[n+3,3], {n,0,30}] (* G. C. Greubel, Jul 20 2019 *)

PARI

Vec(1/(1-4*x)^4+O(x^30)) \\ Charles R Greathouse IV, Oct 03 2016

Sage

[lucas_number2(n, 4, 0)*binomial(n,3)/2^6 for n in range(3, 33)] # Zerinvary Lajos, Mar 11 2009

Formula

a(n) = binomial(n+3, 3)*4^n.

G.f.: 1/(1-4*x)^4.

a(n) = Sum_{a+b+c+d+e+f+g+h=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g)*f(h) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004

From Amiram Eldar, Jan 05 2022: (Start)

Sum_{n>=0} 1/a(n) = 108*log(4/3) - 30.

Sum_{n>=0} (-1)^n/a(n) = 300*log(5/4) - 66. (End)

E.g.f.: exp(4*x)*(3 + 36*x + 72*x^2 + 32*x^3)/3. - Stefano Spezia, Jan 01 2023