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A038208
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Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
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16
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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
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OFFSET
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0,2
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COMMENTS
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Triangle obtained from expansion of (2+2x)^n.
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LINKS
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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.
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.
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FORMULA
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E.g.f. for column k: 2^k*x^k/k!*exp(2*x). - Geoffrey Critzer, Feb 13 2014
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n, k)*2^n, ", "))) \\ ~~~
(MAGMA) [[Binomial(n, k)*2^n: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Oct 17 2018
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CROSSREFS
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Row sums : A000302 (powers of 4).
Sequence in context: A317011 A316876 A317604 * A240484 A240636 A281344
Adjacent sequences: A038205 A038206 A038207 * A038209 A038210 A038211
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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