%I #51 Jan 01 2023 14:52:38
%S 1,16,160,1280,8960,57344,344064,1966080,10813440,57671680,299892736,
%T 1526726656,7633633280,37580963840,182536110080,876173328384,
%U 4161823309824,19585050869760,91396904058880,423311976693760,1947235092791296,8901646138474496,40462027902156800
%N 4-fold convolution of A000302 (powers of 4); expansion of g.f. 1/(1-4*x)^4.
%C 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
%C Also convolution of A020918 with A000984 (central binomial coefficients).
%C 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
%C 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
%C From A152818. a(n) = A006044/6. - _Paul Curtz_, Jan 07 2009
%C Also convolution of A000302 with A038845, also convolution of A002457 with A002802, also convolution of A002697. - _Rui Duarte_, Oct 08 2011
%H Vincenzo Librandi, <a href="/A038846/b038846.txt">Table of n, a(n) for n = 0..400</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (16,-96,256,-256).
%F a(n) = binomial(n+3, 3)*4^n.
%F G.f.: 1/(1-4*x)^4.
%F 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
%F From _Amiram Eldar_, Jan 05 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 108*log(4/3) - 30.
%F Sum_{n>=0} (-1)^n/a(n) = 300*log(5/4) - 66. (End)
%F E.g.f.: exp(4*x)*(3 + 36*x + 72*x^2 + 32*x^3)/3. - _Stefano Spezia_, Jan 01 2023
%p seq(seq(binomial(i, j)*4^(i-3), j =i-3), i=3..33); # _Zerinvary Lajos_, Dec 03 2007
%p seq(binomial(n+3,3)*4^n,n=0..30); # _Zerinvary Lajos_, May 19 2008
%t Table[4^n*Binomial[n+3,3], {n,0,30}] (* _G. C. Greubel_, Jul 20 2019 *)
%o (Sage) [lucas_number2(n, 4, 0)*binomial(n,3)/2^6 for n in range(3, 33)] # _Zerinvary Lajos_, Mar 11 2009
%o (Magma) [4^n*Binomial(n+3, 3): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%o (PARI) Vec(1/(1-4*x)^4+O(x^30)) \\ _Charles R Greathouse IV_, Oct 03 2016
%o (GAP) List([0..30], n-> 4^n*Binomial(n+3,3) ) # _G. C. Greubel_, Jul 20 2019
%Y Cf. A000302, A000984, A002457, A002697, A002802, A006044, A020918, A038231, A038845, A057524, A152818.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_