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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

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Last modified October 23 12:19 EDT 2019. Contains 328345 sequences. (Running on oeis4.)