A238858 -id:A238858 - cp's OEIS frontend

A242153 Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 16, 20, 12, 4, 1, 0, 61, 80, 50, 20, 5, 1, 0, 271, 366, 240, 100, 30, 6, 1, 0, 1372, 1897, 1281, 560, 175, 42, 7, 1, 0, 7795, 10976, 7588, 3416, 1120, 280, 56, 8, 1, 0, 49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, May 05 2014

In general, column k is asymptotic to Pi^(2*k-5/2) / (k! * 6^(k-2) * sqrt(3) * exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

			Triangle T(n,k) begins:
00:      1;
01:      1,     0;
02:      1,     1,     0;
03:      2,     2,     1,     0;
04:      5,     6,     3,     1,    0;
05:     16,    20,    12,     4,    1,    0;
06:     61,    80,    50,    20,    5,    1,   0;
07:    271,   366,   240,   100,   30,    6,   1,  0;
08:   1372,  1897,  1281,   560,  175,   42,   7,  1, 0;
09:   7795, 10976,  7588,  3416, 1120,  280,  56,  8, 1, 0;
10:  49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0;
...
The 15 ascent sequences of length 4 (dots denote zeros) with their number of flat steps are:
01:  [ . . . . ]   3
02:  [ . . . 1 ]   2
03:  [ . . 1 . ]   1
04:  [ . . 1 1 ]   2
05:  [ . . 1 2 ]   1
06:  [ . 1 . . ]   1
07:  [ . 1 . 1 ]   0
08:  [ . 1 . 2 ]   0
09:  [ . 1 1 . ]   1
10:  [ . 1 1 1 ]   2
11:  [ . 1 1 2 ]   1
12:  [ . 1 2 . ]   0
13:  [ . 1 2 1 ]   0
14:  [ . 1 2 2 ]   1
15:  [ . 1 2 3 ]   0
There are 5 sequences without flat steps, 6 with one flat step, etc., giving row [5, 6, 3, 1, 0] for n=4.

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A138265, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163.
Row sums give A022493.
T(2n,n) gives A242164.
Main diagonal and lower diagonals give: A000007, A000012, A000027(n+1), A002378(n+1), A134481(n+1), A130810(n+4).
Cf. A137251 (the same for ascents), A238858 (the same for descents).

b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
      `if`(j=i, x, 1) *b(n-1, j, t+`if`(j>i, 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);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[j == i, x, 1]*b[n-1, j, t + If[j>i, 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, after Alois P. Heinz *)

A066810 Expansion of x^2/((1-3x)(1-2*x)^2).

Original entry on oeis.org

0, 0, 1, 7, 33, 131, 473, 1611, 5281, 16867, 52905, 163835, 502769, 1532883, 4651897, 14070379, 42456897, 127894979, 384799049, 1156756443, 3475250065, 10436235955, 31330727961, 94038321227, 282211432673, 846835624611, 2540926304233, 7623651327931, 22872765923121

Offset: 0

N. J. A. Sloane, Jan 25 2002

Binomial transform of A000295.
a(n) = A112626(n, 2). - Ross La Haye, Jan 11 2006
Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all x,y of P(A), xQy if x is a proper subset of y and |y| - |x| > 1. Then a(n) = |Q|. - Ross La Haye, Jan 11 2008
a(n) is the number of n-digit ternary sequences that have at least two 0's. - Geoffrey Critzer, Apr 14 2009

Harry J. Smith, Table of n, a(n) for n = 0..200
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-16,12).

Column k=1 of A238858 (with different offset).

List([0..30], n-> 3^n - 2^n - n*2^(n-1)); # G. C. Greubel, Nov 18 2019

[3^n-2^n-n*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Nov 29 2015

seq(3^n - 2^n - n*2^(n-1), n=0..30); # G. C. Greubel, Nov 18 2019

RecurrenceTable[{a[n]==3*a[n-1] + (n-1) 2^(n-2), a[0]==0}, a, {n, 0, 30}] (* Geoffrey Critzer, Apr 14 2009 *)
CoefficientList[Series[x^2/((1-3x)(1-2x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 29 2015 *)

a(n) = 3^n -2^n -n*2^(n-1) \\ Harry J. Smith, Mar 29 2010

[3^n - 2^n - n*2^(n-1) for n in (0..30)] # G. C. Greubel, Nov 18 2019

a(n) = 3^n - 2^n - n*2^(n-1).
From Ross La Haye, Apr 26 2006: (Start)
a(n) = A000244(n) - A001792(n).
a(n) = Sum_{k=2..n} binomial(n,k)2^(n-k). (End)
Inverse binomial transform of A086443. - Ross La Haye, Apr 29 2006
Convolution of A000244 beginning [0,1,3,9,27,81,...] and A001787. - Ross La Haye, Feb 15 2007
From Geoffrey Critzer, Apr 14 2009: (Start)
E.g.f.: exp(2*x)*(exp(x) - x - 1).
a(n) = 3*a(n-1) + (n-1)*2^(n-2). (End)

Additional comments from Ross La Haye, Sep 27 2005

A242352 Number T(n,k) of isoscent sequences of length n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=n+2-ceiling(2*sqrt(n+1)), read by rows.

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 6, 21, 29, 2, 51, 124, 28, 127, 499, 241, 10, 323, 1933, 1667, 216, 1, 835, 7307, 10142, 2765, 98, 2188, 27166, 56748, 27214, 2637, 22, 5798, 99841, 299485, 227847, 44051, 1546, 2, 15511, 363980, 1514445, 1708700, 563444, 46947, 570

Offset: 0

Joerg Arndt and Alois P. Heinz, May 11 2014

An isoscent sequence of length n is an integer sequence [s(1),...,s(n)] with s(1) = 0 and 0 <= s(i) <= 1 plus the number of level steps in [s(1),...,s(i)].
Columns k=0-10 give: A001006, A243474, A243475, A243476, A243477, A243478, A243479, A243480, A243481, A243482, A243483.
Row sums give A000110.
Last elements of rows give A243484.

			T(4,0) = 9: [0,0,0,0], [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,1], [0,0,1,2], [0,0,2,2], [0,1,1,1], [0,1,1,2].
T(4,1) = 6: [0,0,1,0], [0,0,2,0], [0,0,2,1], [0,1,0,0], [0,1,0,1], [0,1,1,0].
T(5,2) = 2: [0,0,2,1,0], [0,1,0,1,0].
Triangle T(n,k) begins:
:    1;
:    1;
:    2;
:    4,     1;
:    9,     6;
:   21,    29,     2;
:   51,   124,    28;
:  127,   499,   241,    10;
:  323,  1933,  1667,   216,    1;
:  835,  7307, 10142,  2765,   98;
: 2188, 27166, 56748, 27214, 2637, 22;

Joerg Arndt and Alois P. Heinz, Rows n = 0..114, flattened

Cf. A048993 (for counting level steps), A242351 (for counting ascents), A137251 (ascent sequences counting ascents), A238858 (ascent sequences counting descents), A242153 (ascent sequences counting level steps), A083479.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[jJean-François Alcover, Feb 09 2015, after Maple *)

A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 11, 3, 1, 26, 25, 1, 57, 128, 17, 1, 120, 525, 229, 2, 1, 247, 1901, 1819, 172, 1, 502, 6371, 11172, 3048, 53, 1, 1013, 20291, 58847, 33065, 2751, 7, 1, 2036, 62407, 280158, 275641, 56905, 1422, 1, 4083, 187272, 1242859, 1945529, 771451, 61966, 436

Offset: 0

Joerg Arndt and Alois P. Heinz, May 11 2014

An isoscent sequence of length n is an integer sequence [s(1),...,s(n)] with s(1) = 0 and 0 <= s(i) <= 1 plus the number of level steps in [s(1),...,s(i)].
Columns k=0-10 give: A000012, A000295, A243228, A243229, A243230, A243231, A243232, A243233, A243234, A243235, A243236.
Row sums give A000110.
Last elements of rows give A243237.

			T(4,0) = 1: [0,0,0,0].
T(4,1) = 11: [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,0], [0,0,1,1], [0,0,2,0], [0,0,2,1], [0,0,2,2], [0,1,0,0], [0,1,1,0], [0,1,1,1].
T(4,2) = 3: [0,0,1,2], [0,1,0,1], [0,1,1,2].
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   11,     3;
  1,   26,    25;
  1,   57,   128,    17;
  1,  120,   525,   229,     2;
  1,  247,  1901,  1819,   172;
  1,  502,  6371, 11172,  3048,   53;
  1, 1013, 20291, 58847, 33065, 2751, 7;
  ...

Joerg Arndt and Alois P. Heinz, Rows n = 0..100, flattened

Cf. A048993 (for counting level steps), A242352 (for counting descents), A137251 (ascent sequences counting ascents), A238858 (ascent sequences counting descents), A242153 (ascent sequences counting level steps), A083479.

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
      `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[j>i, x, 1]*b[n-1, j, t + If[j == i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 09 2015, after Maple *)

A241871 Number of ascent sequences of length 2n with exactly n descents.

Original entry on oeis.org

1, 0, 0, 1, 26, 937, 45747, 2945144, 242899690, 25034354941, 3157647587689, 478931493603308, 86057396214591300, 18086772915953351382, 4397414569504319733812, 1224945090342466220614714, 387654163770235904289085798, 138333762956844287480268151988

Offset: 0

Joerg Arndt and Alois P. Heinz, May 01 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..70

Cf. A238858.

b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
      `if`(ji, 1, 0)), j=0..t+1)))
    end:
a:= n-> coeff(b(2*n, -1$2), x, n):
seq(a(n), n=0..20);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[ji, 1, 0]], {j, 0, t+1}]]]; a[n_] := Coefficient[b[2*n, -1, -1], x, n]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 13 2015, after Maple *)

a(n) = A238858(2n,n).

A241872 Number of ascent sequences of length n with exactly two descents.

Original entry on oeis.org

4, 53, 429, 2748, 15342, 78339, 376159, 1728458, 7689744, 33393393, 142376385, 598555320, 2489143090, 10264270175, 42048021027, 171366151974, 695585112660, 2814484154445, 11359684937605, 45759869226260, 184050366838134, 739376299832763, 2967455421451239

Offset: 5

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (17,-121,467,-1054,1388,-984,288).

Column k=2 of A238858.

gf := -(12*x^2-15*x+4)*x^5/((4*x-1)*(x-1)*(3*x-1)^2*(2*x-1)^3):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=5..30);

CoefficientList[Series[-(12 x^2 - 15 x + 4)/((4 x - 1) (x - 1) (3 x - 1)^2 (2 x - 1)^3), {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2014 *)
LinearRecurrence[{17,-121,467,-1054,1388,-984,288},{4,53,429,2748,15342,78339,376159},23] (* Ray Chandler, Jul 14 2015 *)

G.f.: -(12*x^2-15*x+4)*x^5/((4*x-1)*(x-1)*(3*x-1)^2*(2*x-1)^3).
a(n) = 4^n/6 - 3^(n-1)*(2*n+1)/4 + 2^(n-4)*(n+2)*(n-1) + 1/12. - Vaclav Kotesovec, May 03 2014
Recurrence: a(n) = 288*a(n-7) - 984*a(n-6) + 1388*a(n-5) - 1054*a(n-4) + 467*a(n-3) - 121*a(n-2) + 17*a(n-1). - Fung Lam, May 05 2014

A241873 Number of ascent sequences of length n with exactly three descents.

Original entry on oeis.org

1, 48, 822, 9305, 83590, 647891, 4537169, 29532566, 182034751, 1076357002, 6162251432, 34394051095, 188121970788, 1012370499109, 5376927101387, 28254655805724, 147182871736245, 761235618312420, 3914066453608570, 20027841005048805, 102071452026321906

Offset: 6

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (32,-461,3952,-22443,88896,-251663,512656,-745096,752672,-500976,196992,-34560).

Column k=3 of A238858.

gf:= -(912*x^6-2440*x^5+2481*x^4-1177*x^3+253*x^2-16*x-1)*x^6/
      ((5*x-1)*(4*x-1)^2*(x-1)^2*(3*x-1)^3*(2*x-1)^4):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=6..30);

CoefficientList[Series[-(912 x^6 - 2440 x^5 + 2481 x^4 - 1177 x^3 + 253 x^2 - 16 x - 1)/((5 x - 1) (4 x - 1)^2 (x - 1)^2 (3 x - 1)^3 (2 x - 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2014 *)
LinearRecurrence[{32,-461,3952,-22443,88896,-251663,512656,-745096,752672,-500976,196992,-34560},{1,48,822,9305,83590,647891,4537169,29532566,182034751,1076357002,6162251432,34394051095},21] (* Ray Chandler, Jul 14 2015 *)

G.f.: -(912*x^6-2440*x^5+2481*x^4-1177*x^3+253*x^2-16*x-1)*x^6 / ((5*x-1) *(4*x-1)^2 *(x-1)^2 *(3*x-1)^3 *(2*x-1)^4).
a(n) = 3*5^(n-1)/8 - 4^(n-1)*n/3 + 3^(n-2)*(6*n^2-2*n-7)/16 - 2^(n-5)*(n-2)*(n-1)*(n+3)/3 - n/24 + 1/16. - Vaclav Kotesovec, May 03 2014
Recurrence: a(n) = -34560*a(n-12) + 196992*a(n-11) - 500976*a(n-10) + 752672*a(n-9) - 745096*a(n-8) + 512656*a(n-7) - 251663*a(n-6) + 88896*a(n-5) - 22443*a(n-4) + 3952*a(n-3) - 461*a(n-2) + 32*a(n-1). - Fung Lam, May 05 2014

A241874 Number of ascent sequences of length n with exactly four descents.

Original entry on oeis.org

26, 1048, 21362, 307660, 3574869, 35900857, 324623778, 2713846040, 21359949568, 160346402882, 1159100542422, 8127076433744, 55581620321035, 372410647606299, 2453173612562310, 15932182184914620, 102249194946464430, 649680545407603980, 4093272335570479850

Offset: 8

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature(53, -1308, 19968, -211254, 1644366, -9756556, 45104276, -164641137, 477888789, -1105173264, 2030572644, -2940614560, 3309217712, -2828604672, 1770962112, -764322048, 202798080, -24883200).

Column k=4 of A238858.

gf:= -(466560*x^10 -1981728*x^9 +3631752*x^8 -3741230*x^7 +2365035*x^6 -936340*x^5 +223475*x^4 -27090*x^3 +174*x^2 +330*x -26) *x^8 / ((6*x-1) *(5*x-1)^2 *(4*x-1)^3 *(x-1)^3 *(3*x-1)^4 *(2*x-1)^5):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=8..30);

CoefficientList[Series[-(466560 x^10 - 1981728 x^9 + 3631752 x^8 - 3741230 x^7 + 2365035 x^6 - 936340 x^5 + 223475 x^4 - 27090 x^3 + 174 x^2 + 330 x - 26)/((6 x - 1) (5 x - 1)^2 (4 x - 1)^3 (x - 1)^3 (3 x - 1)^4 (2 x - 1)^5), {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2014 *)
LinearRecurrence[{53, -1308, 19968, -211254, 1644366, -9756556, 45104276, -164641137, 477888789, -1105173264, 2030572644, -2940614560, 3309217712, -2828604672, 1770962112, -764322048, 202798080, -24883200},{26, 1048, 21362, 307660, 3574869, 35900857, 324623778, 2713846040, 21359949568, 160346402882, 1159100542422, 8127076433744, 55581620321035, 372410647606299, 2453173612562310, 15932182184914620, 102249194946464430, 649680545407603980}, 20];(* Ray Chandler, Jul 15 2015 *)

G.f.: -(466560*x^10 -1981728*x^9 +3631752*x^8 -3741230*x^7 +2365035*x^6 -936340*x^5 +223475*x^4 -27090*x^3 +174*x^2 +330*x -26) *x^8 / ((6*x-1) *(5*x-1)^2 *(4*x-1)^3 *(x-1)^3 *(3*x-1)^4 *(2*x-1)^5).

Recurrence: a(n) = - 24883200*a(n-18) + 202798080*a(n-17) - 764322048*a(n-16) + 1770962112*a(n-15) - 2828604672*a(n-14) + 3309217712*a(n-13) - 2940614560*a(n-12) + 2030572644*a(n-11) - 1105173264*a(n-10) + 477888789*a(n-9) - 164641137*a(n-8) + 45104276*a(n-7) - 9756556*a(n-6) + 1644366*a(n-5) - 211254*a(n-4) + 19968*a(n-3) - 1308*a(n-2) + 53*a(n-1). - Fung Lam, May 05 2014

A241875 Number of ascent sequences of length n with exactly five descents.

Original entry on oeis.org

8, 937, 35841, 834115, 14475124, 206587438, 2564426795, 28685171369, 296140258017, 2869968329846, 26436819147050, 233659504323986, 1995996397796603, 16573895612885901, 134389245968036082, 1068038768762441634, 8344630999626596958, 64255565358244018191

Offset: 9

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=5 of A238858.

gf:= -(1789516800*x^16 -10641300480*x^15 +28799118720*x^14 -46828104768*x^13 +50870603132*x^12 -38817852028*x^11 +21260210219*x^10 -8353318594*x^9 +2286195777*x^8 -394486012*x^7 +25511399*x^6 +6059293*x^5 -1915919*x^4 +243868*x^3 -15144*x^2 +289*x+8) *x^9 / ((7*x-1) *(6*x-1)^2 *(5*x-1)^3 *(4*x-1)^4 *(x-1)^4 *(3*x-1)^5 *(2*x-1)^6):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=9..30);

CoefficientList[Series[-(1789516800 x^16 - 10641300480 x^15 + 28799118720 x^14 - 46828104768 x^13 + 50870603132 x^12 - 38817852028 x^11 + 21260210219 x^10 - 8353318594 * x^9 + 2286195777 * x^8 - 394486012 * x^7 + 25511399 x^6 + 6059293 x^5 - 1915919 x^4 + 243868 x^3 - 15144 x^2 + 289 x + 8)/((7 x - 1) (6 x - 1)^2 (5 x - 1)^3 (4 x - 1)^4 (x - 1)^4 (3 x - 1)^5 (2 x - 1)^6), {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2014 *)

G.f.: see Maple program.

Recurrence: a(n) = 81*a(n-1) - 3114*a(n-2) + 75618*a(n-3) - 1302183*a(n-4) + 16924743*a(n-5) - 172522788*a(n-6) + 1414869228*a(n-7) - 9501687423*a(n-8) + 52906702383*a(n-9) - 246402134298*a(n-10) + 965539475298*a(n-11) - 3194875953273*a(n-12) + 8941122759033*a(n-13) - 21157696301688*a(n-14) + 42243068089008*a(n-15) - 70868692309248*a(n-16) + 99257760429408*a(n-17) - 114988409883008*a(n-18) + 108762502457088*a(n-19) - 82478130147072*a(n-20) + 48857688836352*a(n-21) - 21745388335104*a(n-22) + 6829114613760*a(n-23) - 1347275980800*a(n-24) + 125411328000*a(n-25). - Fung Lam, May 05 2014

A241876 Number of ascent sequences of length n with exactly six descents.

Original entry on oeis.org

1, 594, 45747, 1752513, 45552389, 920513763, 15577569349, 231095209005, 3098219351061, 38346553035796, 445033714399778, 4900020726869918, 51649070462238267, 524892382085986515, 5172330086955870408, 49648755377072570286, 465988523060678103585

Offset: 10

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=6 of A238858.

gf:= -(57480192000000*x^23 -445890272256000*x^22+1619423860193280*x^21 -3652404812826624*x^20 +5721231909570048*x^19 -6594644113079904*x^18 +5779639610824024*x^17 -3921044317402316*x^16 +2072911198114226*x^15 -849438228561495*x^14 +263329934846856*x^13 -57503572316263*x^12
+6610760803436*x^11 +737845435920*x^10 -563506749299*x^9 +146999655366*x^8 -24201211392*x^7 +2687528742*x^6 -192088566*x^5 +6983684*x^4 +69774*x^3 -17175*x^2 +477*x+1) *x^10 / ((8*x-1) *(7*x-1)^2 *(6*x-1)^3 *(5*x-1)^4 *(4*x-1)^5 *(x-1)^5 *(3*x-1)^6 *(2*x-1)^7):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=10..30);

G.f.: see Maple program.
Recurrence:
a(n) = 5056584744960000*a(n-33) - 68065242513408000*a(n-32) + 437803880015462400*a(n-31) - 1792719557254840320*a(n-30) + 5253036459825954816*a(n-29) - 11738495444617580544*a(n-28) + 20817456370349202432*a(n-27) - 30104829351014344704*a(n-26) + 36199103820290982144*a(n-25) - 36720645434814626560*a(n-24) + 31774115324102749056*a(n-23) - 23653530457275554304*a(n-22) + 15249527617274617248*a(n-21) - 8558081781076074864*a(n-20) + 4196991567155864940*a(n-19) - 1803694897653672920*a(n-18) + 680558169447265365*a(n-17) - 225674624312836065*a(n-16) + 65779879124624040*a(n-15) - 16842763995507000*a(n-14) + 3782553545656620*a(n-13) - 743262373360860*a(n-12) + 127338118210800*a(n-11) - 18930771191160*a(n-10) + 2426885862174*a(n-9) - 266102398566*a(n-8) + 24690049848*a(n-7) - 1911283016*a(n-6) + 121101516*a(n-5) - 6114540*a(n-4) + 236484*a(n-3) - 6576*a(n-2) + 117*a(n-1). - Fung Lam, May 06 2014

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A242153 Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A242352 Number T(n,k) of isoscent sequences of length n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=n+2-ceiling(2*sqrt(n+1)), read by rows.

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A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows.

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A241871 Number of ascent sequences of length 2n with exactly n descents.

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A241872 Number of ascent sequences of length n with exactly two descents.

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A241873 Number of ascent sequences of length n with exactly three descents.

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A066810 Expansion of x^2/((1-3x)(1-2*x)^2).