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A038207
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Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
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52
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1, 2, 1, 4, 4, 1, 8, 12, 6, 1, 16, 32, 24, 8, 1, 32, 80, 80, 40, 10, 1, 64, 192, 240, 160, 60, 12, 1, 128, 448, 672, 560, 280, 84, 14, 1, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 1024, 5120, 11520, 15360, 13440
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| This infinite matrix is the square of the Pascal matrix (A007318) whose rows are [ 1,0,... ], [ 1,1,0,... ], [ 1,2,1,0,... ],...
As an upper right triangle, table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley (se16(AT)btinternet.com), Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber (christof.weber(AT)fhnw.ch), May 08 2009
Number of different partial sums of 1+[1,1,2]+[2,2,3]+[3,3,4]+[4,4,5]+... with entries that are zero removed. - Jon Perry (perry(AT)globalnet.co.uk), Jan 01 2004
Row sums are powers of 3 (A000244), antidiagonal sums are Pell numbers (A000129). - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), May 17 2005
Riordan array (1/(1-2x),x/(1-2x)). - Paul Barry (pbarry(AT)wit.ie), Jul 28 2005
T(n,k) is the number of elements of the Coxeter group B_n with descent set contained in {s_k}, 0<=k<=n-1. For T(n,n), we interpret this as the number of elements of B_n with empty descent set (since s_n does not exist). - Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006
Let S be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xSy if x is a subset of y. Then T(n,k) = the number of elements (x,y) of S for which y has exactly k more elements than x. - Ross La Haye (rlahaye(AT)new.rr.com), Oct 12 2007
T(n,k) is number of paths in the first quadrant going from (0,0) to (n,k) using only steps B=(1,0) colored blue, R=(1,0) colored red and U=(1,1). Example: T(3,2)=6 because we have BUU, RUU, UBU, URU, UUB and UUR. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 2007
T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (0,1), and two kinds of step (1,0). [Joerg Arndt, Jul 01 2011]
T(i,j) is the number of i-permutations of {1,2,3} containing j 1's. Example: T(2,1)=4 because we have 12, 13, 21 and 31; T(3,2)=6 because we have 112, 113, 121, 131, 211 and 311. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 21 2007
Triangle of coefficients in expansion of (2+x)^n. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Apr 13 2008
Sum of diagonals are Jacobsthal-numbers: A001045 [From M. Dols (markdols99(AT)yahoo.com), Aug 31 2009]
Triangle T(n,k), read by rows, given by [2,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 15 2009]
Eigensequence of the triangle = A004211: (1, 3, 11, 49, 257, 1539,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 07 2010]
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 155.
H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
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LINKS
| T. D. Noe, Rows n=0..100 of triangle, flattened
John Cartan, Starmaze: Cartan's Triangle.
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FORMULA
| T(n, k) = sum(i=0..n, C(n, i)*C(i, k) ).
G.f.: 1/(1-2*z-t*z). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 2007
Rows of the triangle are generated by taking successive iterates of (A135387)^n * [1, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 09 2007
From the formalism of A133314, the e.g.f. for the row polynomials of A038207 is exp(x*t)*exp(2x). The e.g.f. for the row polynomials of the inverse matrix is exp(x*t)*exp(-2x). p iterates of the matrix give the matrix with e.g.f. exp(x*t)*exp(p*2x). The results generalize for 2 replaced by any number. [From Tom Copeland (tcjpn(AT)msn.com), Aug 18 2008]
sum(0<=k<=n, T(n,k)*x^k ) = (2+x)^n. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 15 2009]
n-th row is obtained by taking pairwise sums of triangle A112857 terms starting from the right. - Gary W. Adamson, Feb 06 2012
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EXAMPLE
| Triangle begins:
1
2...1
4...4...1
8...12..6...1
16..32..24..8..1
32..80..80..40..10..1
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MAPLE
| for i from 0 to 12 do seq(binomial(i, j)*2^(i-j), j = 0 .. i) end do; # yields sequence in t riangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 04 2007
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MATHEMATICA
| Table[CoefficientList[Expand[(y + x + x^2)^n], y] /. x -> 1, {n, 0, 10}] // TableForm (*Geoffrey Critzer, Nov 20 2011*)
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PROG
| (PARI) T(n, k)=polcoeff((x+2)^n, k) - Michael Somos, Apr 27 2000
(PARI) { n=13; v=vector(n); for (i=1, n, v[i]=vector(3^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+i; v[i][j+k]=v[i-1][j]+i; v[i][j+k+k]=v[i-1][j]+i+1)); c=vector(n); for (i=1, n, for (j=1, 3^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } (Jon Perry)
(Haskell)
a038207 n = a038207_list !! n
a038207_list = concat $ iterate ([2, 1] *) [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
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CROSSREFS
| Cf. A007318, A013609, A013610, etc. See also A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A062715.
Cf. A065109, A135387.
Apart from signs, same as A065109.
A004211 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 07 2010]
Cf. A112857
Sequence in context: A134397 A134395 * A065109 A113988 A134308 A202710
Adjacent sequences: A038204 A038205 A038206 * A038208 A038209 A038210
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KEYWORD
| nonn,tabl,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Example corrected by Clark Kimberling, Aug 5 2011.
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