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A013609
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Triangle of coefficients in expansion of (1+2*x)^n.
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26
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1, 1, 2, 1, 4, 4, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 1, 20, 180, 960
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and two kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. [Joerg Arndt, Jul 01 2011]
Also sum of rows in A046816. - Lior Manor (lior.manor(AT)gmail.com) Apr 24 2004
Also square array of unsigned coefficients of Chebyshev polynomials of second kind . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 12 2005
The rows give the number of k-simplices in the n-cube. For example, 1, 6, 12, 8 shows that the 3-cube has 1 volume, 6 faces, 12 edges and 8 vertices. - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jun 05 2006
Triangle whose (i, j)-th entry is binomial(i, j)*2^j.
With offset [1,1] the triangle with doubled numbers, 2*a(n,m), enumerates sequences of length m with nonzero integer entries n_i satisfying sum(|n_i|)<=n. Example n=4, m=2: [1,3], [3,1], [2,2] each in 2^2=4 signed versions: 2*a(4,2)=2*6=12. The Sum over m (row sums of 2*a(n,m)) gives 2*3^(n-1), n>=1. See the W. Lang comment and a K. A. Meissner reference under A024023. - W. Lang, Jan 21 2008.
n-th row of the triangle = leftmost column of nonzero terms of X^n, where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (2,2,2,...) in the subdiagonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008
Numerators of a matrix square-root of Pascal's triangle A007318, where the denominators for the n-th row are set to 2^n. [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Aug 20 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
The triangle sums, see A180662 for their definitions, link the Pell-Jacobsthal triangle, its mirror image is A038207, with twenty-four different sequences, see the crossrefs.
This triangle may very well be called the Pell-Jacobsthal triangle in view of the fact that A000129 (Kn21) are the Pell numbers and A001045 (Kn11) the Jacobsthal numbers.
(End)
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REFERENCES
| B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
John Cartan, Cartan's triangle shows the relationship to the n-cube.
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FORMULA
| G.f.: 1 / (1 - x*(1+2*y)).
T(n,k) = 2^k*binomial(n,k).
bin2(n, k) = 2*bin2(n-1, k-1) + bin2(n-1, k) (e.g. 1, 4, 4 gives 1, 2.1+4=6, 2.4+4=8 and 2.4=8) - Jon Perry (perry(AT)globalnet.co.uk), Nov 22 2005
Row sums are 3^n = A000244(n). [Joerg Arndt, Jul 01 2011]
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EXAMPLE
| Triangle begins:
1;
1, 2;
1, 4, 4;
1, 6, 12, 8;
1, 8, 24, 32, 16;
1, 10, 40, 80, 80, 32;
1, 12, 60, 160, 240, 192, 64;
1, 14, 84, 280, 560, 672, 448, 128;
1, 16, 112, 448, 1120, 1792, 1792, 1024, 256;
1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512;
1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024;
1, 22, 220, 1320, 5280, 14784, 29568, 42240, 42240, 28160, 11264, 2048;
1, 24, 264, 1760, 7920, 25344, 59136, 101376, 126720, 112640, 67584, 24576, 4096;
1, 26, 312, 2288, 11440, 41184, 109824, 219648, 329472, 366080, 292864, 159744, 53248, 8192;
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MAPLE
| bin2:=proc(n, k) option remember; if k<0 or k>n then 0 elif k=0 then 1 else 2*bin2(n-1, k-1)+bin2(n-1, k); fi; end; # N. J. A. Sloane, Jun 01 2009
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MATHEMATICA
| Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* From Jean-François Alcover, May 17 2011 *)
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PROG
| (Haskell)
a013609 n = a013609_list !! n
a013609_list = concat $ iterate ([1, 2] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
(p:ps) + (q:qs) = p + q : ps + qs
ps + qs = ps ++ qs
(p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
_ * _ = []
-- Reinhard Zumkeller, Apr 02 2011
(PARI) /* same as in A092566 but use */
steps=[[1, 0], [1, 1], [1, 1]]; /* note double [1, 1] */
/* Joerg Arndt, Jul 01 2011 */
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CROSSREFS
| Cf. A007318, A013610, etc.
Appears in A167580 and A167591. [Johannes W. Meijer, Nov 23 2009]
From Johannes W. Meijer (meijgia(AT)hotmail.com), Sep 22 2010: (Start)
Triangle sums (see the comments): A000244 (Row1); A000012 (Row2); A001045 (Kn11); A026644 (Kn12); 4*A011377 (Kn13); A000129 (Kn21); A094706 (Kn22); A099625 (Kn23); A001653 (Kn3); A007583 (Kn4); A046717 (Fi1); A007051 (Fi2); A077949 (Ca1); A008998 (Ca2); A180675 (Ca3); A092467 (Ca4); A052942 (Gi1); A008999 (Gi2); A180676 (Gi3); A180677 (Gi4); A140413 (Ze1); A180678 (Ze2); A097117 (Ze3); A055588 (Ze4).
(End)
Sequence in context: A136672 A097750 A133544 * A154558 A008572 A118976
Adjacent sequences: A013606 A013607 A013608 * A013610 A013611 A013612
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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