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 A238858 Triangle T(n,k) read by rows: T(n,k) is the number of length-n ascent sequences with exactly k descents. 16
 1, 1, 0, 2, 0, 0, 4, 1, 0, 0, 8, 7, 0, 0, 0, 16, 33, 4, 0, 0, 0, 32, 131, 53, 1, 0, 0, 0, 64, 473, 429, 48, 0, 0, 0, 0, 128, 1611, 2748, 822, 26, 0, 0, 0, 0, 256, 5281, 15342, 9305, 1048, 8, 0, 0, 0, 0, 512, 16867, 78339, 83590, 21362, 937, 1, 0, 0, 0, 0, 1024, 52905, 376159, 647891, 307660, 35841, 594, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Columns k=0-10 give: A011782, A066810(n-1), A241872, A241873, A241874, A241875, A241876, A241877, A241878, A241879, A241880. The sequence of column k satisfies a linear recurrence with constant coefficients of order k*(k+5)/2 = A055998(k) for k>0. T(2n,n) gives A241871(n). Last nonzero elements of rows give A241881(n). Row sums give A022493. LINKS Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE Triangle starts: 00:     1; 01:     1,      0; 02:     2,      0,       0; 03:     4,      1,       0,       0; 04:     8,      7,       0,       0,       0; 05:    16,     33,       4,       0,       0,      0; 06:    32,    131,      53,       1,       0,      0,     0; 07:    64,    473,     429,      48,       0,      0,     0,   0; 08:   128,   1611,    2748,     822,      26,      0,     0,   0, 0; 09:   256,   5281,   15342,    9305,    1048,      8,     0,   0, 0, 0; 10:   512,  16867,   78339,   83590,   21362,    937,     1,   0, 0, 0, 0; 11:  1024,  52905,  376159,  647891,  307660,  35841,   594,   0, 0, 0, 0, 0; 12:  2048, 163835, 1728458, 4537169, 3574869, 834115, 45747, 262, 0, 0, 0, 0, 0; ... The 53 ascent sequences of length 5 together with their numbers of descents are (dots for zeros): 01:  [ . . . . . ]   0      28:  [ . 1 1 . 1 ]   1 02:  [ . . . . 1 ]   0      29:  [ . 1 1 . 2 ]   1 03:  [ . . . 1 . ]   1      30:  [ . 1 1 1 . ]   1 04:  [ . . . 1 1 ]   0      31:  [ . 1 1 1 1 ]   0 05:  [ . . . 1 2 ]   0      32:  [ . 1 1 1 2 ]   0 06:  [ . . 1 . . ]   1      33:  [ . 1 1 2 . ]   1 07:  [ . . 1 . 1 ]   1      34:  [ . 1 1 2 1 ]   1 08:  [ . . 1 . 2 ]   1      35:  [ . 1 1 2 2 ]   0 09:  [ . . 1 1 . ]   1      36:  [ . 1 1 2 3 ]   0 10:  [ . . 1 1 1 ]   0      37:  [ . 1 2 . . ]   1 11:  [ . . 1 1 2 ]   0      38:  [ . 1 2 . 1 ]   1 12:  [ . . 1 2 . ]   1      39:  [ . 1 2 . 2 ]   1 13:  [ . . 1 2 1 ]   1      40:  [ . 1 2 . 3 ]   1 14:  [ . . 1 2 2 ]   0      41:  [ . 1 2 1 . ]   2 15:  [ . . 1 2 3 ]   0      42:  [ . 1 2 1 1 ]   1 16:  [ . 1 . . . ]   1      43:  [ . 1 2 1 2 ]   1 17:  [ . 1 . . 1 ]   1      44:  [ . 1 2 1 3 ]   1 18:  [ . 1 . . 2 ]   1      45:  [ . 1 2 2 . ]   1 19:  [ . 1 . 1 . ]   2      46:  [ . 1 2 2 1 ]   1 20:  [ . 1 . 1 1 ]   1      47:  [ . 1 2 2 2 ]   0 21:  [ . 1 . 1 2 ]   1      48:  [ . 1 2 2 3 ]   0 22:  [ . 1 . 1 3 ]   1      49:  [ . 1 2 3 . ]   1 23:  [ . 1 . 2 . ]   2      50:  [ . 1 2 3 1 ]   1 24:  [ . 1 . 2 1 ]   2      51:  [ . 1 2 3 2 ]   1 25:  [ . 1 . 2 2 ]   1      52:  [ . 1 2 3 3 ]   0 26:  [ . 1 . 2 3 ]   1      53:  [ . 1 2 3 4 ]   0 27:  [ . 1 1 . . ]   1 There are 16 ascent sequences with no descent, 33 with one, and 4 with 2, giving row 4 [16, 33, 4, 0, 0, 0]. MAPLE # b(n, i, t): polynomial in x where the coefficient of x^k is   # #             the number of postfixes of these sequences of     # #             length n having k descents such that the prefix   # #             has rightmost element i and exactly t ascents     # b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(       `if`(ji, 1, 0)), j=0..t+1)))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, -1\$2)): seq(T(n), n=0..12); MATHEMATICA b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[If[ji, 1, 0]], {j, 0, t+1}]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, -1, -1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *) PROG (Sage) # Transcription of the Maple program R. = QQ[] @CachedFunction def b(n, i, t):     if n==0: return 1     return sum( ( x if ji) ) for j in range(t+2) ) def T(n): return b(n, -1, -1) for n in range(0, 10): print(T(n).list()) CROSSREFS Cf. A137251 (ascent sequences with k ascents), A242153 (ascent sequences with k flat steps). Sequence in context: A273346 A136334 A155039 * A106235 A288098 A118965 Adjacent sequences:  A238855 A238856 A238857 * A238859 A238860 A238861 KEYWORD nonn,tabl,look AUTHOR Joerg Arndt and Alois P. Heinz, Mar 06 2014 STATUS approved

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Last modified September 22 14:14 EDT 2020. Contains 337291 sequences. (Running on oeis4.)