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A038208 Triangle whose (i,j)-th entry is binomial(i,j)*2^i.

%I A038208 #39 Mar 17 2025 11:26:34
%S A038208 1,2,2,4,8,4,8,24,24,8,16,64,96,64,16,32,160,320,320,160,32,64,384,
%T A038208 960,1280,960,384,64,128,896,2688,4480,4480,2688,896,128,256,2048,
%U A038208 7168,14336,17920,14336,7168,2048,256,512,4608,18432,43008,64512,64512,43008,18432,4608,512
%N A038208 Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
%C A038208 Triangle obtained from expansion of (2 + 2*x)^n.
%H A038208 Harvey P. Dale, <a href="/A038208/b038208.txt">Table of n, a(n) for n = 0..1000</a>
%H A038208 B. N. Cyvin et al., <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match34/match34_109-121.pdf">Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons</a>, Match, No. 34 (Oct 1996), pp. 109-121.
%H A038208 S. Elizalde, E. Deutsch, <a href="https://doi.org/10.54550/ECA2022V2S1R7">The degree of asymmetry of a sequence</a>, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, corollary 2.2 at m=2, then dropping each 2nd row
%H A038208 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H A038208 Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H A038208 Franck Ramaharo, <a href="https://arxiv.org/abs/2002.06672">A bracket polynomial for 2-tangle shadows</a>, arXiv:2002.06672 [math.CO], 2020.
%F A038208 E.g.f. for column k: 2^k*x^k/k!*exp(2*x). - _Geoffrey Critzer_, Feb 13 2014
%F A038208 From _G. C. Greubel_, Mar 21 2022: (Start)
%F A038208 T(n, n-k) = T(n, k).
%F A038208 T(n, 0) = A000079(n).
%F A038208 Sum_{k=0..n} T(n, k) = A000302(n).
%F A038208 Sum_{k=0..floor(n/2)} T(n-k, k) = A002605(n+1).
%F A038208 Sum_{k=0..floor(n/2)} T(n, k) = 2^n*A027306(n). (End)
%e A038208     1;
%e A038208     2,    2;
%e A038208     4,    8,    4;
%e A038208     8,   24,   24,     8;
%e A038208    16,   64,   96,    64,    16;
%e A038208    32,  160,  320,   320,   160,   32;
%e A038208    64,  384,  960,  1280,   960,   384,   64;
%e A038208   128,  896, 2688,  4480,  4480,  2688,  896,  128;
%e A038208   256, 2048, 7168, 14336, 17920, 14336, 7168, 2048, 256;
%t A038208 nn=8;Map[Select[#,#>0&]&,Transpose[Table[Range[0,nn]!CoefficientList[Series[2^k x^k/k! Exp[2x],{x,0,nn}],x],{k,0,nn}]]]//Grid (* _Geoffrey Critzer_, Feb 13 2014 *)
%t A038208 Flatten[Table[Binomial[i,j]2^i,{i,0,10},{j,0,i}]] (* _Harvey P. Dale_, May 28 2015 *)
%o A038208 (PARI) for(n=0,10, for(k=0,n, print1(binomial(n,k)*2^n, ", "))) \\ _G. C. Greubel_, Oct 17 2018
%o A038208 (Magma) [Binomial(n,k)*2^n: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Oct 17 2018
%o A038208 (Sage) flatten([[binomial(n,k)*2^n for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 21 2022
%Y A038208 Cf. A000079, A000302 (row sums), A002605 (diagonal sums), A027306.
%K A038208 nonn,tabl,easy
%O A038208 0,2
%A A038208 _N. J. A. Sloane_