The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A241701 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A241701

Number T(n,k) of Carlitz compositions of n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.
(history; published version)

#21 by Alois P. Heinz at Sun Apr 30 09:49:38 EDT 2017

STATUS

editing

approved

#20 by Alois P. Heinz at Sun Apr 30 09:41:30 EDT 2017

FORMULA

Sum_{k=0..floor(n/3)} (k+1) * T(n,k) = A285994(n) (for n>0).

CROSSREFS

Cf. A238344, A285994.

STATUS

approved

editing

#19 by Alois P. Heinz at Fri Sep 04 10:46:08 EDT 2015

STATUS

editing

approved

#18 by Alois P. Heinz at Fri Sep 04 10:46:04 EDT 2015

NAME

Number T(n,k) of Carlitz compositions of n with k exactly k descents; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

STATUS

approved

editing

#17 by Bruno Berselli at Fri Feb 13 06:02:07 EST 2015

STATUS

proposed

approved

#16 by Jean-François Alcover at Fri Feb 13 04:58:34 EST 2015

STATUS

editing

proposed

#15 by Jean-François Alcover at Fri Feb 13 04:58:28 EST 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[If[j == i, 0, b[n-j, j]*If[j<i, x, 1]], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

STATUS

approved

editing

#14 by Alois P. Heinz at Sun Apr 27 13:24:15 EDT 2014

STATUS

editing

approved

#13 by Alois P. Heinz at Sun Apr 27 12:21:54 EDT 2014

NAME

Number T(n,k) of Carlitz compositions of n with k exactly descents; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

#12 by Alois P. Heinz at Sun Apr 27 12:04:43 EDT 2014

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, expand(

add(`if`(j=i, 0, b(n-j, j)*`if`(j<i, x, 1)), j=1..n)))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):

seq(T(n), n=0..20);