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A225624 Triangle read by rows: T(n,k) is the number of descent sequences of length n with exactly k-1 descents, n>=1, 1<=k<=n. 2
1, 2, 0, 3, 1, 0, 4, 5, 0, 0, 5, 15, 3, 0, 0, 6, 35, 25, 1, 0, 0, 7, 70, 117, 28, 0, 0, 0, 8, 126, 405, 271, 22, 0, 0, 0, 9, 210, 1155, 1631, 483, 13, 0, 0, 0, 10, 330, 2871, 7359, 5126, 711, 5, 0, 0, 0, 11, 495, 6435, 27223, 36526, 13482, 889, 1, 0, 0, 0, 12, 715, 13299, 86919, 199924, 151276, 30906, 962, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A descent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= 1 + desc([d(1), d(2), ..., d(k-1)]) where desc(.) gives the number of descents of its argument, see example.

Row sums are A225588 (number of descent sequences).

First column is C(n,1)=n, second column is C(n+1,4) = A000332(n+1), third column appears to be A095664(n-5) for n>=5.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 1..100, flattened (Rows n = 1..18 from Joerg Arndt)

EXAMPLE

Triangle begins:

01:  1,

02:  2, 0,

03:  3, 1, 0,

04:  4, 5, 0, 0,

05:  5, 15, 3, 0, 0,

06:  6, 35, 25, 1, 0, 0,

07:  7, 70, 117, 28, 0, 0, 0,

08:  8, 126, 405, 271, 22, 0, 0, 0,

09:  9, 210, 1155, 1631, 483, 13, 0, 0, 0,

10:  10, 330, 2871, 7359, 5126, 711, 5, 0, 0, 0,

11:  11, 495, 6435, 27223, 36526, 13482, 889, 1, 0, 0, 0,

12:  12, 715, 13299, 86919, 199924, 151276, 30906, 962, 0, 0, 0, 0,

13:  13, 1001, 25740, 247508, 903511, 1216203, 546001, 63462, 903, 0, 0, 0, 0,

...

The number of descents for the A225588(5)=23 descent sequences of length 5 are (dots for zeros):

.#:  descent seq.   no. of descents

01:  [ . . . . . ]    0

02:  [ . . . . 1 ]    0

03:  [ . . . 1 . ]    1

04:  [ . . . 1 1 ]    0

05:  [ . . 1 . . ]    1

06:  [ . . 1 . 1 ]    1

07:  [ . . 1 . 2 ]    1

08:  [ . . 1 1 . ]    1

09:  [ . . 1 1 1 ]    0

10:  [ . 1 . . . ]    1

11:  [ . 1 . . 1 ]    1

12:  [ . 1 . . 2 ]    1

13:  [ . 1 . 1 . ]    2

14:  [ . 1 . 1 1 ]    1

15:  [ . 1 . 1 2 ]    1

16:  [ . 1 . 2 . ]    2

17:  [ . 1 . 2 1 ]    2

18:  [ . 1 . 2 2 ]    1

19:  [ . 1 1 . . ]    1

20:  [ . 1 1 . 1 ]    1

21:  [ . 1 1 . 2 ]    1

22:  [ . 1 1 1 . ]    1

23:  [ . 1 1 1 1 ]    0

There are 5 sequences with 0 descents, 15 with 1 descents, 3 with 2 descents, and 0 for 3 or 5 descents. Therefore row 5 is [5, 15, 3, 0, 0].

MAPLE

b:= proc(n, i, t) option remember; local j; if n<1 then [0$t, 1]

      else []; for j from 0 to t+1 do zip((x, y)->x+y, %,

      b(n-1, j, t+`if`(j<i, 1, 0)), 0) od; % fi

    end:

T:= proc(n) local l; l:= b(n-1, 0, 0): l[], 0$(n-nops(l)) end:

seq(T(n), n=1..13);  # Alois P. Heinz, May 18 2013

MATHEMATICA

b[n_, i_, t_] :=  b[n, i, t] =  Module[{j, pc}, If[n<1, Append[Array[0 &, t], 1], pc = {}; For[j = 0, j <= t+1, j++, pc = Plus @@ PadRight[ {pc, b[n-1, j, t+If[j<i, 1, 0]]}]]; pc]]; T[n_] := Module[{l}, l = b[n-1, 0, 0]; Join[l, Array[0&, n-Length[l]]]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 27 2014, after Alois P. Heinz *)

PROG

(Sage)  # After Alois P. Heinz.

@CachedFunction

def b(n, i, t, N):

    B = [0 for x in range(N)]

    if n < 1: B[t] = 1; return B

    for j in (0..t+1):

        B = map(operator.add, B, b(n-1, j, t+int(j<i), N))

    return B

def T(n): return b(n-1, 0, 0, n)

for n in (1..9): T(n)  #  Peter Luschny, May 20 2013; updated May 21 2013

CROSSREFS

Sequence in context: A284871 A202064 A144955 * A168020 A225084 A238345

Adjacent sequences:  A225621 A225622 A225623 * A225625 A225626 A225627

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, May 11 2013

STATUS

approved

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Last modified May 23 12:45 EDT 2017. Contains 286915 sequences.