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

0,2

Comments

Triangle obtained from expansion of (2 + 2*x)^n.

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.

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.

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)

Example

    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;

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 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n, k)*2^n, ", "))) \\ G. C. Greubel, Oct 17 2018
(Magma) [Binomial(n, k)*2^n: k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2018
(Sage) flatten([[binomial(n, k)*2^n for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

Crossrefs

Cf. A000079, A000302 (row sums), A002605 (diagonal sums), A027306.

Sequence in context: A317011 A316876 A317604 * A240484 A240636 A281344

Adjacent sequences: A038205 A038206 A038207 * A038209 A038210 A038211

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane

Status

approved