A038208 Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
1, 2, 2, 4, 8, 4, 8, 24, 24, 8, 16, 64, 96, 64, 16, 32, 160, 320, 320, 160, 32, 64, 384, 960, 1280, 960, 384, 64, 128, 896, 2688, 4480, 4480, 2688, 896, 128, 256, 2048, 7168, 14336, 17920, 14336, 7168, 2048, 256, 512, 4608, 18432, 43008, 64512, 64512, 43008, 18432, 4608, 512
Offset: 0
Examples
1;
2, 2;
4, 8, 4;
8, 24, 24, 8;
16, 64, 96, 64, 16;
32, 160, 320, 320, 160, 32;
64, 384, 960, 1280, 960, 384, 64;
128, 896, 2688, 4480, 4480, 2688, 896, 128;
256, 2048, 7168, 14336, 17920, 14336, 7168, 2048, 256;
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
- S. Elizalde, E. Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, corollary 2.2 at m=2, then dropping each 2nd row
- Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
- Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
- Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.
Programs
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Magma
[Binomial(n,k)*2^n: k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2018
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Mathematica
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 *) Flatten[Table[Binomial[i,j]2^i,{i,0,10},{j,0,i}]] (* Harvey P. Dale, May 28 2015 *) -
PARI
for(n=0,10, for(k=0,n, print1(binomial(n,k)*2^n, ", "))) \\ G. C. Greubel, Oct 17 2018
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Sage
flatten([[binomial(n,k)*2^n for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022
Formula
E.g.f. for column k: 2^k*x^k/k!*exp(2*x). - Geoffrey Critzer, Feb 13 2014
From G. C. Greubel, Mar 21 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
Sum_{k=0..n} T(n, k) = A000302(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A002605(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = 2^n*A027306(n). (End)
Comments