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A155038 Triangle read by rows: T(n,k) is the number of compositions of n with first part k. 4
1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, 32, 16, 8, 4, 2, 1, 1, 64, 32, 16, 8, 4, 2, 1, 1, 128, 64, 32, 16, 8, 4, 2, 1, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 2048, 1024, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Previous name was: Matrix inverse of A154990.

Apart from first term essentially the same as A057728.

A011782 appears in the columns.

Riordan array ((1-x)/(1-2x), x). - Philippe Deléham, Jan 24 2010

Indexing the triangle from n=0 and k=0, T(n,k) is the number of binary words of length n that begin with a run of exactly k 0's. O.g.f.: 1/((1-y*x)*(1-x/(1-x))). - Geoffrey Critzer, Feb 15 2012

LINKS

Reinhard Zumkeller, Rows n = 1..100 of table, flattened

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.

FORMULA

T(j,k) = A011782(j-k), j>=1, k>=1. - Omar E. Pol, Feb 14 2013

T(n,k) = 2^{n-k-1} if k<n; T(n,n) = 1; T(n,k) = 0 if k>n. - Emeric Deutsch, Jan 12 2018

G.f.:  G(t,x) = (1-2x+tx^2)/(1-2x)(1-tx)). - Emeric Deutsch, Jan 19 2018

EXAMPLE

T(5,2) = 4 because the compositions of 5 with first part 2 are: [2,3], [2,2,1], [2,1,2], and [2,1,1,1]. - Emeric Deutsch, Jan 12 2018

Table begins:

   1,

   1,  1,

   2,  1,  1,

   4,  2,  1,  1,

   8,  4,  2,  1,  1,

  16,  8,  4,  2,  1,  1,

  32, 16,  8,  4,  2,  1,  1,

  64, 32, 16,  8,  4,  2,  1,  1,

Production matrix begins:

  1, 1

  1, 0, 1

  1, 0, 0, 1

  1, 0, 0, 0, 1

  1, 0, 0, 0, 0, 1

  1, 0, 0, 0, 0, 0, 1

  1, 0, 0, 0, 0, 0, 0, 1

  1, 0, 0, 0, 0, 0, 0, 0, 1

  ... - Philippe Deléham, Oct 04 2014

MAPLE

T := proc(n, k) if k = n then 1 elif k < n then 2^(n-k-1) else 0 end if end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 12 2018

G:= (1-2*x+t*x^2)/((1-2*x)*(1-t*x)): Gser := simplify(series(G, x = 0, 15)): for n to 14 do P[n] := coeff(Gser, x, n) end do: for n to 14 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 19 2018

MATHEMATICA

nn = 15; a = 1/(1 - y x); f[list_] := Select[list, # > 0 &]; Map[f, CoefficientList[Series[ a/(1 - x/(1 - x)), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Feb 15 2012 *)

PROG

(Haskell)

a155038 n k = a155038_tabl !! (n-1) !! (k-1)

a155038_row n = a155038_tabl !! (n-1)

a155038_tabl = iterate

   (\row -> zipWith (+) (row ++ [0]) (init row ++ [0, 1])) [1]

-- Reinhard Zumkeller, Aug 08 2013

CROSSREFS

Cf. A011782, A057728, A154990, A155033, A155039.

Sequence in context: A140996 A141020 A152568 * A057728 A176463 A098050

Adjacent sequences:  A155035 A155036 A155037 * A155039 A155040 A155041

KEYWORD

nonn,tabl

AUTHOR

Mats Granvik, Jan 19 2009

EXTENSIONS

New name from Joerg Arndt, May 04 2014

STATUS

approved

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Last modified January 20 14:26 EST 2020. Contains 331094 sequences. (Running on oeis4.)