A242153 -id:A242153 - cp's OEIS frontend

A138265 Number of upper triangular zero-one matrices with n ones and no zero rows or columns.

1, 1, 1, 2, 5, 16, 61, 271, 1372, 7795, 49093, 339386, 2554596, 20794982, 182010945, 1704439030, 17003262470, 180011279335, 2015683264820, 23801055350435, 295563725628564, 3850618520827590, 52514066450469255, 748191494586458700, 11115833059268126770

Offset: 0

Vladeta Jovovic, Mar 10 2008, Mar 11 2008

Row sums of A193357.
This is also the number of rigid unlabeled interval orders with n points (see Brightwell-Keller, Theorem 2; or Dukes-Kitaev-Remmel-Steingrímsson, Theorem 8). - N. J. A. Sloane, Dec 04 2011  [Corrected by Vít Jelínek, Sep 04 2014.]
Number of length-n ascent sequences without flat steps (i.e., no two adjacent digits are equal).  An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(k)>=0 and d(k) <= 1 + asc([d(1), d(2), ..., d(k-1)]) and asc(.) gives the number of ascents of its argument. [Joerg Arndt, Nov 05 2012]

			From _Joerg Arndt_, Nov 05 2012: (Start)
The a(4) = 5 such matrices with 4 ones are (dots for zeros):
  1 . . .      1 1 .      1 . 1      1 1 .      1 . .
  . 1 . .      . . 1      . 1 .      . 1 .      . 1 1
  . . 1 .      . . 1      . . 1      . . 1      . . 1
  . . . 1
The a(5)=16 ascent sequences without flat steps are (dots for zeros):
  [ 1]   [ . 1 . 1 . ]
  [ 2]   [ . 1 . 1 2 ]
  [ 3]   [ . 1 . 1 3 ]
  [ 4]   [ . 1 . 2 . ]
  [ 5]   [ . 1 . 2 1 ]
  [ 6]   [ . 1 . 2 3 ]
  [ 7]   [ . 1 2 . 1 ]
  [ 8]   [ . 1 2 . 2 ]
  [ 9]   [ . 1 2 . 3 ]
  [10]   [ . 1 2 1 . ]
  [11]   [ . 1 2 1 2 ]
  [12]   [ . 1 2 1 3 ]
  [13]   [ . 1 2 3 . ]
  [14]   [ . 1 2 3 1 ]
  [15]   [ . 1 2 3 2 ]
  [16]   [ . 1 2 3 4 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
G. E. Andrews and J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345 [math.NT], 2014 (see final paragraph of text).
George E. Andrews and Vít Jelínek, On q-Series Identities Related to Interval Orders, arXiv:1309.6669 [math.CO], 2013.
George E. Andrews and Vít Jelínek, On q-Series Identities Related to Interval Orders, European Journal of Combinatorics, Volume 39, July 2014, 178-187.
Graham Brightwell and Mitchel T. Keller, Asymptotic Enumeration of Labelled Interval Orders, arXiv 1111.6766 [math.CO], 2011.
Kathrin Bringmann, Yingkun Li, and Robert C. Rhoades, Asymptotics for the number of row-Fishburn matrices, European Journal of Combinatorics, vol.41, pp.183-196, (October-2014); preprint.
Matthieu Dien, Antoine Genitrini, and Frederic Peschanski, A Combinatorial Study of Async/Await Processes, Conf.: 19th Int'l Colloq. Theor. Aspects of Comp. (2022), (Analytic) Combinatorics of concurrent systems.
M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, arXiv:1006.2696 [math.CO], 2010.
M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, Journal of Combinatorics, Volume 2, Issue 1, 2011, pp. 139-163.
F. Garvan, Congruences and relations for the r-Fishburn numbers, arXiv:1406.5611 [math.NT], 2014.
Ankush Goswami, Abhash Kumar Jha, Byungchan Kim, and Robert Osburn, Asymptotics and sign patterns for coefficients in expansions of Habiro elements, arXiv:2204.02628 [math.NT], 2022.
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
V. Jelínek, Counting self-dual interval orders, arXiv:1106.2261 [math.CO], 2011.
V. Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614.
V. Jelinek, Catalan pairs and Fishburn triples, Adv. Appl. Math. 70 (2015) 1-31
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order (journal), published on-line, 2011.
Yan X. Zhang, Four Variations on Graded Posets, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

Cf. A005321, A104602, A135588, A193548, A194530.
Column k=0 of A242153.
Column k=1 of A264909.
Row sums of A137252.

g:=sum(product(1-1/(1+x)^i,i=1..n),n=0..35): gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=0..22);  # Emeric Deutsch, Mar 23 2008
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n<1, 1, add(
     `if`(i=j, 0, b(n-1, j, t+`if`(j>i, 1, 0))), j=0..t+1))
    end:
a:= n-> b(n-1, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 09 2012, Jan 14 2015

max = 25; g = Sum[Product[1 - 1/(1 - x)^i, {i, 1, n}], {n, 0, max}]; gser = Series[g, {x, 0, max}]; a[n_] := SeriesCoefficient[gser, {x, 0, n}]; Table[a[n] // Abs, {n, 0, max-1}] (* Jean-François Alcover, Jan 24 2014, after Emeric Deutsch *)

# Adaptation of the Maple program by Alois P. Heinz:
@CachedFunction
def b(n, i, t):
    if n<1: return 1
    return sum(b(n-1, j, t+(j>i)) for j in range(t+2))
def a(n):
    if n<1: return 1
    return sum((-1)^(n-k)*binomial(n-1, k-1)*b(k-1, 0, 0) for k in range(n+1))
[a(n) for n in range(33)]
# Joerg Arndt, Feb 26 2014

G.f.: Sum_{n>=0} (Product_{i=1..n} 1-1/(1+x)^i).
G.f.: Sum_{n>=0} (1+x)^(n+1)*Product_{i=1..n} (1-(1+x)^i)^2. Proved by Bringmann-Li-Rhoades, and by Andrews-Jelínek. - Vít Jelínek, Sep 04 2014
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A079144(k). a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*A022493(k).
G.f.: B(x/(1+x)) where B(x) is the g.f. of A022493; g.f.: Q(0,u) where u=x/(1+x), Q(k,u) = 1 + (1 - (1-x)^(2*k+1))/(1 - (1-(1-x)^(2*k+2))/(1 -(1-x)^(2*k+2) + 1/Q(k+1,u) )); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
Asymptotics (Brightwell and Keller, 2011): a(n) ~ 12*sqrt(3)/(exp(Pi^2/12)*Pi^(5/2)) * n!*sqrt(n)*(6/Pi^2)^n. - Vaclav Kotesovec, May 03 2014
From Vít Jelínek, Sep 04 2014: (Start)
For each m, a(5m+4) mod 5 = 0. Conjectured by Andrews-Sellers, and proved by Garvan (see Remark 1.4(ii) in Garvan's paper).
For each m, a(5m+1) mod 5 = a(5m+2) mod 5 = 3*a(5m+3) mod 5. Proved by Garvan (see (1.17) in Garvan's paper).
The limit a(n)/A022493(n) is equal to exp(-Pi^2/6). This corresponds to the asymptotic probability that a random unlabeled interval order is rigid (See Brightwell-Keller; or Jelínek, Fact 5.2). (End)
Conjectural g.f.:  1 + Sum_{n >= 0} n/(1+x)^(n+1) * (Product_{i = 1..n} 1 - 1/(1+x)^i). Cf. A194530. - Peter Bala, Aug 21 2023

More terms from Emeric Deutsch, Mar 23 2008

A238858 Triangle T(n,k) read by rows: T(n,k) is the number of length-n ascent sequences with exactly k descents.

Original entry on oeis.org

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

Offset: 0

Joerg Arndt and Alois P. Heinz, Mar 06 2014

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.

			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].

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

Cf. A137251 (ascent sequences with k ascents), A242153 (ascent sequences with k flat steps).

# 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);

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 *)

# 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())

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 *)

A242154 Number of ascent sequences of length n with exactly one flat step.

Original entry on oeis.org

1, 2, 6, 20, 80, 366, 1897, 10976, 70155, 490930, 3733246, 30655152, 270334766, 2548153230, 25566585450, 272052199520, 3060191748695, 36282298766760, 452220051658265, 5911274512571280, 80862988937379390, 1155309461910323610, 17208404375488550100

Offset: 2

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 2..140

Column k=1 of A242153.

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}]]]; a[n_] := Coefficient[b[n, -1, -1], x, 1]; Table[ a[n], {n, 2, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ 2*sqrt(3)/(exp(Pi^2/12)*sqrt(Pi)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242155 Number of ascent sequences of length n with exactly two flat steps.

Original entry on oeis.org

1, 3, 12, 50, 240, 1281, 7588, 49392, 350775, 2700115, 22399476, 199258488, 1892343362, 19111149225, 204532683600, 2312443695920, 27541725738255, 344681838284220, 4522200516582650, 62068382381998440, 889492878311173290, 13286058811968721515

Offset: 3

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 3..140

Column k=2 of A242153.

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}]]]; a[n_] := Coefficient[b[n, -1, -1], x, 2]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(3/2)/(2*sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242156 Number of ascent sequences of length n with exactly three flat steps.

Original entry on oeis.org

1, 4, 20, 100, 560, 3416, 22764, 164640, 1286175, 10800460, 97064396, 929872944, 9461716810, 101926129200, 1159018540400, 13874662175520, 174430929675615, 2297878921894800, 31655403616078550, 455168137467988560, 6819445400385661890, 106288470495749772120

Offset: 4

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 4..140

Column k=3 of A242153.

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}]]]; a[n_] := Coefficient[b[n, -1, -1], x, 3]; Table[a[n], {n, 4, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(7/2)/(36*sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242157 Number of ascent sequences of length n with exactly four flat steps.

Original entry on oeis.org

1, 5, 30, 175, 1120, 7686, 56910, 452760, 3858525, 35101495, 339725386, 3487023540, 37846867240, 433186049100, 5215583431800, 65904645333720, 872154648378075, 12063864339947700, 174104719888432025, 2617216790440934220, 40916672402313971340

Offset: 5

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 5..140

Column k=4 of A242153.

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}]]]; a[n_] := Coefficient[b[n, -1, -1], x, 4]; Table[a[n], {n, 5, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(11/2)/(864*sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242158 Number of ascent sequences of length n with exactly five flat steps.

Original entry on oeis.org

1, 6, 42, 280, 2016, 15372, 125202, 1086624, 10032165, 98284186, 1019176158, 11158475328, 128679348616, 1559469776760, 19819217040840, 263618581334880, 3663049523187915, 53081003095769880, 800881711486787315, 12562640594116484256, 204583362011569856700

Offset: 6

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 6..140

Column k=5 of A242153.

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}]]]; a[n_] := Coefficient[b[n, -1, -1], x, 5]; Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(15/2)/(25920*sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

A242159 Number of ascent sequences of length n with exactly six flat steps.

Original entry on oeis.org

1, 7, 56, 420, 3360, 28182, 250404, 2354352, 23408385, 245710465, 2717803088, 31615680096, 386038045848, 4938320959740, 66064056802800, 922665034672080, 13431181585022355, 203477178533784540, 3203526845947149260, 52344335808818684400, 886527902050136045700

Offset: 7

Joerg Arndt and Alois P. Heinz, May 05 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 7..140

Column k=6 of A242153.

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}]]]; a[n_] := Coefficient[b[n, -1, -1], x, 6]; Table[a[n], {n, 7, 30}] (* Jean-François Alcover, Feb 10 2015, after A242153 *)

a(n) ~ Pi^(19/2) / (6! * 6^4 * sqrt(3)*exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - Vaclav Kotesovec, Aug 27 2014

cp's OEIS Frontend

A138265 Number of upper triangular zero-one matrices with n ones and no zero rows or columns.

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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.

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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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A242154 Number of ascent sequences of length n with exactly one flat step.

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A242155 Number of ascent sequences of length n with exactly two flat steps.

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A242156 Number of ascent sequences of length n with exactly three flat steps.

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A242157 Number of ascent sequences of length n with exactly four flat steps.

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