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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
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-François 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 A321878 A225084
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, May 11 2013
STATUS
approved