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 A013609 Triangle of coefficients in expansion of (1+2*x)^n. 53
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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, Apr 24 2004 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 From Johannes W. Meijer, Sep 22 2010: (Start) 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 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. G. Hotz, Zur Reduktion von Schaltkreispolynomen im Hinblick auf eine Verwendung in Rechenautomaten, El. Datenverarbeitung, Folge 5 (1960), pp. 21-27. 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. J. Goldman, 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. W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021. D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35. 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). - Jon Perry, Nov 22 2005 Row sums are 3^n = A000244(n). - Joerg Arndt, Jul 01 2011 T(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i). - Mircea Merca, Apr 28 2012 E.g.f.: exp(2*y*x + x). - Geoffrey Critzer, Nov 12 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 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; From Peter Bala, Apr 20 2012: (Start) 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) 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 MATHEMATICA Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* Jean-François Alcover, May 17 2011 *) 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 (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] -- Reinhard Zumkeller, Jul 22 2013, Feb 27 2013 (PARI) /* same as in A092566 but use */ steps=[[1, 0], [1, 1], [1, 1]]; /* note double [1, 1] */ /* Joerg Arndt, Jul 01 2011 */ (Maxima) a(n, k):=coeff(expand((1+2*x)^n), x^k); create_list(a(n, k), n, 0, 6, k, 0, n); \\ Emanuele Munarini, Nov 21 2012 CROSSREFS Cf. A007318, A013610, etc. Appears in A167580 and A167591. - Johannes W. Meijer, Nov 23 2009 From Johannes W. Meijer, 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) Cf. A105728, A115068. Sequence in context: A304623 A133544 A303872 * A154558 A220836 A008572 Adjacent sequences:  A013606 A013607 A013608 * A013610 A013611 A013612 KEYWORD tabl,nonn,easy,nice AUTHOR STATUS approved

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Last modified October 17 05:23 EDT 2018. Contains 316275 sequences. (Running on oeis4.)