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A013609
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Triangle of coefficients in expansion of (1+2*x)^n.
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57
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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, 3360, 8064, 13440, 15360, 11520, 5120, 1024
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OFFSET
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0,3
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COMMENTS
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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 square array of unsigned coefficients of Chebyshev polynomials of second kind. - Philippe Deléham, 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, 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. - Wolfdieter 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, 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. - Gerald McGarvey, Aug 20 2009
The triangle sums (see A180662 for their definitions) link the Pell-Jacobsthal triangle, whose 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)
T(n,k) equals the number of n-length words on {0,1,2} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 2-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
T(n,k) is the number of chains 0=x_0<x_1<...<x_k=1 in the butterfly poset of rank n. Cf. Ehrenborg and Readdy link. - Geoffrey Critzer, Oct 01 2022
Excluding the initial 1, T(n,k) is the number of k-faces of a regular n-cross polytope. See A038207 for n-cube and A135278 for n-simplex. - Mohammed Yaseen, Jan 14 2023
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REFERENCES
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B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
G. Hotz, Zur Reduktion von Schaltkreispolynomen im Hinblick auf eine Verwendung in Rechenautomaten, El. Datenverarbeitung, Folge 5 (1960), pp. 21-27.
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LINKS
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H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients of k rooks on the 2xn board, all heights 2.
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FORMULA
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G.f.: 1 / (1 - x*(1+2*y)).
T(n,k) = 2^k*binomial(n,k).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k). - Jon Perry, Nov 22 2005
T(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i). - Mircea Merca, Apr 28 2012
Riordan array (x/(1 - x)), 2*x/(1 - x)). Exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(1 + 6*x + 12*x^2/2! + 8*x^3/3!) = 1 + 8*x + 40*x^2/2! + 160*x^3/3! + 560*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 2*x/(1 - x) ). - Peter Bala, Dec 21 2014
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 3^j. - Kolosov Petro, Jan 28 2019
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EXAMPLE
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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;
The triangle can be written as the matrix product A038207*(signed version of A013609).
|.1................||.1..................|
|.2...1............||-1...2..............|
|.4...4...1........||.1..-4...4..........|
|.8..12...6...1....||-1...6...-12...8....|
|16..32..24...8...1||.1..-8....24.-32..16|
|..................||....................|
(End)
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MAPLE
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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
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Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* Jean-François Alcover, May 17 2011 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 3], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)
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PROG
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(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
_ * _ = []
(Haskell)
a013609 n k = a013609_tabl !! n !! k
a013609_row n = a013609_tabl !! n
a013609_tabl = iterate (\row -> zipWith (+) ([0] ++ row) $
zipWith (+) ([0] ++ row) (row ++ [0])) [1]
(PARI) /* same as in A092566 but use */
steps=[[1, 0], [1, 1], [1, 1]]; /* note double [1, 1] */
(Maxima) a(n, k):=coeff(expand((1+2*x)^n), x^k);
(Magma) [2^k*Binomial(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2021
(Sage) flatten([[2^k*binomial(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 17 2021
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CROSSREFS
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(End)
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KEYWORD
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AUTHOR
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STATUS
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approved
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