A155038 Triangle read by rows: T(n,k) is the number of compositions of n with first part k.
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
Offset: 1
Examples
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
Links
- Reinhard Zumkeller, Rows n = 1..100 of table, flattened
- Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
- Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices and Riordan arrays, arXiv:math/0702638 [math.CO], 2007.
Programs
-
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
-
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 *)
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 kn. - Emeric Deutsch, Jan 12 2018
G.f.: G(t,x) = (1-2*x+t*x^2)/((1-2*x)*(1-t*x)). - Emeric Deutsch, Jan 19 2018
Extensions
New name from Joerg Arndt, May 04 2014
Comments