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%I #58 Mar 23 2024 20:02:56
%S 1,12,96,640,3840,21504,114688,589824,2949120,14417920,69206016,
%T 327155712,1526726656,7046430720,32212254720,146028888064,
%U 657129996288,2937757630464,13056700579840,57724360458240,253987186016256
%N 3-fold convolution of A000302 (powers of 4).
%C Also convolution of A002802 with A000984 (central binomial coefficients).
%C With a different offset, number of n-permutations of 5 objects u, v, w, z, x with repetition allowed, containing exactly two u's. - _Zerinvary Lajos_, Dec 29 2007
%C Also convolution of A000302 with A002697, also convolution of A002457 with itself. - _Rui Duarte_, Oct 08 2011
%H Vincenzo Librandi, <a href="/A038845/b038845.txt">Table of n, a(n) for n = 0..400</a>
%H Adam Ehrenberg, Joseph T. Iosue, Abhinav Deshpande, Dominik Hangleiter, and Alexey V. Gorshkov, <a href="https://arxiv.org/abs/2403.13878">The Second Moment of Hafnians in Gaussian Boson Sampling</a>, arXiv:2403.13878 [quant-ph], 2024. See p. 30.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,64).
%F a(n) = (n+2)*(n+1)*2^(2*n-1).
%F G.f.: 1/(1-4*x)^3.
%F a(n) = Sum_{a+b+c+d+e+f=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f) with f(n)=A000984(n). - _Philippe Deléham_, Jan 22 2004
%F a(n) = binomial(n+2,n) * 4^n. - _Rui Duarte_, Oct 08 2011
%F E.g.f.: (1 + 8*x + 8*x^2)*exp(4*x). - _G. C. Greubel_, Jul 20 2019
%F From _Amiram Eldar_, Jan 05 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 8 - 24*log(4/3).
%F Sum_{n>=0} (-1)^n/a(n) = 40*log(5/4) - 8. (End)
%p seq((n+2)*(n+1)*4^n/2, n=0..30); # _Zerinvary Lajos_, Apr 25 2007
%t Table[4^n*Binomial[n+2,n], {n,0,30}] (* _G. C. Greubel_, Jul 20 2019 *)
%o (Sage) [4^(n-2)*binomial(n,2) for n in range(2, 30)] # _Zerinvary Lajos_, Mar 11 2009
%o (Magma) [4^n*Binomial(n+2, 2): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011
%o (PARI) a(n)=(n+2)*(n+1)<<(2*n-1) \\ _Charles R Greathouse IV_, Aug 21 2015
%o (GAP) List([0..30], n-> 4^n*Binomial(n+2,n) ); # _G. C. Greubel_, Jul 20 2019
%Y Cf. A000302, A000984, A002802, A038231, A052780.
%Y Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), A027472 (q=3), this sequence (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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