A022493 -id:A022493 - cp's OEIS frontend

A202059 Number of ascent sequences avoiding the pattern 100.

1, 1, 2, 5, 14, 44, 153, 583, 2410, 10721, 50965, 257393, 1374187, 7722862, 45520064, 280502924, 1802060232, 12040040899, 83475921469, 599400745354, 4449689901306, 34096169966924, 269286884243138, 2189193150557825, 18297258191472880, 157049750065028868

Offset: 0

N. J. A. Sloane, Dec 10 2011

nonn

It appears that no formula or g.f. is known.

Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..628
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011

Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317.

Corrected a(7), was 383, but should be 583 according to Duncan-Steimgrimsson paper and independent computation. - Andrew Baxter, Jan 06 2014
a(0) and a(15)-a(21) from Alois P. Heinz, Jan 06 2014
a(22) from Alois P. Heinz, Oct 06 2014
a(23) from Alois P. Heinz, Apr 20 2016
More terms from Anthony Guttmann, Nov 04 2021

A202060 Number of ascent sequences avoiding the pattern 110.

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 143, 510, 1936, 7774, 32848, 145398, 671641, 3227218, 16084747, 82955090, 441773793, 2424845273, 13695855478, 79485625385, 473393639992, 2889930405750, 18064609329598, 115513453404597, 754956282308784, 5039064184597772, 34323984497482559

Offset: 0

N. J. A. Sloane, Dec 10 2011

nonn

It appears that no formula or g.f. is known.

Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..43
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011

Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317.

a(0) and a(15)-a(17) from Alois P. Heinz, Jan 07 2014

More terms from Anthony Guttmann, Nov 04 2021

A202061 Number of ascent sequences avoiding the pattern 120.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 133, 442, 1535, 5546, 20754, 80113, 317875, 1292648, 5374073, 22794182, 98462847, 432498659, 1929221610, 8728815103, 40017844229, 185727603829, 871897549029, 4137132922197, 19828476952117, 95934298966615, 468291607852143, 2305162065138433

Offset: 0

N. J. A. Sloane, Dec 10 2011

nonn

It appears that no formula or g.f. is known.

Liang Chengwei, Shi Lecun and Cai Zhongyu, Table of n, a(n) for n = 0..500 (terms 0..74 from Andrew Conway and Miles Conway)
Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.

Total number of ascent sequences is given by A022493. Number of ascent sequences avoiding 001 (and others) is A000079; 102 is A007051; 101 is A000108; 000 is A202058; 100 is A202059; 110 is A202060; 120 is A202061; 201 is A202062; 210 is A108304; 0123 is A080937; 0021 is A007317.

More terms from Anthony Guttmann, Nov 04 2021

A218580 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with first occurrence of the maximal value at position k-1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 1, 8, 13, 15, 16, 1, 16, 35, 47, 56, 62, 1, 32, 97, 153, 204, 248, 279, 1, 64, 275, 515, 770, 1030, 1257, 1423, 1, 128, 793, 1785, 3000, 4424, 5869, 7140, 8100, 1, 256, 2315, 6347, 12026, 19582, 28293, 37058, 44843, 50887

Offset: 1

Joerg Arndt, Nov 03 2012

Row sums are A022493.

Second column are powers of 2.

			Triangle starts:
[ 1]  1,
[ 2]  1, 1,
[ 3]  1, 2, 2,
[ 4]  1, 4, 5, 5,
[ 5]  1, 8, 13, 15, 16,
[ 6]  1, 16, 35, 47, 56, 62,
[ 7]  1, 32, 97, 153, 204, 248, 279,
[ 8]  1, 64, 275, 515, 770, 1030, 1257, 1423,
[ 9]  1, 128, 793, 1785, 3000, 4424, 5869, 7140, 8100,
[10]  1, 256, 2315, 6347, 12026, 19582, 28293, 37058, 44843, 50887,
...

Cf. A022493 (number of ascent sequences).
Cf. A218579 (ascent sequences with last zero at position k-1), A218581 (ascent sequences with last occurrence of the maximal value at position k-1).
Cf. A137251 (ascent sequences with k ascents), A218577 (ascent sequences with maximal element k), A175579 (ascent sequences with k zeros).

A218581 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with last occurrence of the maximal value at position k-1.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 10, 0, 1, 6, 15, 31, 0, 1, 10, 29, 62, 115, 0, 1, 18, 63, 148, 288, 496, 0, 1, 34, 149, 392, 826, 1496, 2437, 0, 1, 66, 375, 1120, 2592, 5036, 8615, 13435, 0, 1, 130, 989, 3392, 8698, 18332, 33391, 54548, 82127

Offset: 1

Joerg Arndt, Nov 03 2012

Row sums are A022493.

			Triangle starts:
[ 1]  1,
[ 2]  0, 2,
[ 3]  0, 1, 4,
[ 4]  0, 1, 4, 10,
[ 5]  0, 1, 6, 15, 31,
[ 6]  0, 1, 10, 29, 62, 115,
[ 7]  0, 1, 18, 63, 148, 288, 496,
[ 8]  0, 1, 34, 149, 392, 826, 1496, 2437,
[ 9]  0, 1, 66, 375, 1120, 2592, 5036, 8615, 13435,
[10]  0, 1, 130, 989, 3392, 8698, 18332, 33391, 54548, 82127,
[11]  0, 1, 258, 2703, 10768, 30768, 70868, 138635, 239688, 377001, 551384,
...

Cf. A022493 (number of ascent sequences).
Cf. A218579 (ascent sequences with last zero at position k-1), A218580 (ascent sequences with first occurrence of the maximal value at position k-1).
Cf. A137251 (ascent sequences with k ascents), A218577 (ascent sequences with maximal element k), A175579 (ascent sequences with k zeros).

A294220 Number A(n,k) of ascent sequences of length n where no letter multiplicity is larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 10, 1, 0, 1, 1, 2, 5, 14, 27, 1, 0, 1, 1, 2, 5, 15, 47, 83, 1, 0, 1, 1, 2, 5, 15, 52, 180, 277, 1, 0, 1, 1, 2, 5, 15, 53, 210, 773, 1015, 1, 0, 1, 1, 2, 5, 15, 53, 216, 964, 3701, 4007, 1, 0

Offset: 0

Alois P. Heinz, Oct 25 2017

			A(4,2) = 10: 0123, 0011, 0012, 0101, 0102, 0110, 0112, 0120, 0121, 0122.
Square array A(n,k) begins:
  1, 1,    1,    1,    1,    1,    1,    1,    1, ...
  0, 1,    1,    1,    1,    1,    1,    1,    1, ...
  0, 1,    2,    2,    2,    2,    2,    2,    2, ...
  0, 1,    4,    5,    5,    5,    5,    5,    5, ...
  0, 1,   10,   14,   15,   15,   15,   15,   15, ...
  0, 1,   27,   47,   52,   53,   53,   53,   53, ...
  0, 1,   83,  180,  210,  216,  217,  217,  217, ...
  0, 1,  277,  773,  964, 1006, 1013, 1014, 1014, ...
  0, 1, 1015, 3701, 4960, 5270, 5326, 5334, 5335, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv:1109.3641, 2011

Columns k=0-3 give: A000007, A000012, A202058, A317784.
Main diagonal gives A022493.
Cf. A294219.

b:= proc(n, i, t, p, k) option remember; `if`(n=0, 1,
      add(`if`(coeff(p, x, j)=k, 0, b(n-1, j, t+
          `if`(j>i, 1, 0), p+x^j, k)), j=1..t+1))
    end:
A:= (n, k)-> b(n, 0$3, min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n == 0, 1, Sum[ If[ Coefficient[p, x, j] == k, 0, b[n-1, j, t + If[j>i, 1, 0], p + x^j, k]], {j, 1, t+1}]];
A[n_, k_] := b[n, 0, 0, 0, Min[n, k]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Aug 05 2018, translated from Maple *)

A(n,k) = Sum_{j=0..k} A294219(n,j).

A(n,k) = A(n,n) = A022493(n) for k >= n.

A099960 An interleaving of the Genocchi numbers of the first and second kind, A110501 and A005439.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 8, 17, 56, 155, 608, 2073, 9440, 38227, 198272, 929569, 5410688, 28820619, 186043904, 1109652905, 7867739648, 51943281731, 401293838336, 2905151042481, 24290513745920, 191329672483963, 1721379917619200, 14655626154768697, 141174819474169856

Offset: 0

N. J. A. Sloane, Nov 13 2004

First column (also row sums) of triangle in A099959.

Number of ascent sequences of length n without level steps and with alternating ascents and descents. a(6) = 8: 010101, 010102, 010103, 010201, 010202, 010203, 010212, 010213. - Alois P. Heinz, Oct 27 2017

Donald E. Knuth, The Art of Computer Programming, Vol. 4, fascicle 1, section 7.1.4, p. 220, answer to exercise 174, Addison-Wesley, 2009.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Catalin Zara, Cardinality of l_1-Segments and Genocchi Numbers, arXiv:1304.5798 [math.CO] (2013)

Cf. A022493, A099959, A001469, A005439, A138265, A294281.

with(linalg):rev:=proc(a) local n, p; n:=vectdim(a): p:=i->a[n+1-i]: vector(n,p) end: ps:=proc(a) local n, q; n:=vectdim(a): q:=i->sum(a[j],j=1..i): vector(n,q) end: pss:=proc(a) local n, q; n:=vectdim(a): q:=proc(i) if i<=n then sum(a[j],j=1..i) else sum(a[j],j=1..n) fi end: vector(n+1,q) end: R[0]:=vector(1,1): for n from 1 to 30 do if n mod 2 = 1 then R[n]:=ps(rev(R[n-1])) else R[n]:=pss(rev(R[n-1])) fi od: seq(R[n][1],n=0..30); # Emeric Deutsch

g1 = Table[2*(4^n-1)*BernoulliB[2*n] // Abs, {n, 0, 13}]; g2 = Table[2*(-1)^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}], {n, 0, 13}]; Riffle[g1, g2] // Rest (* Jean-François Alcover, May 23 2013 *)

# Algorithm of L. Seidel (1877)
def A099960_list(n) :
    D = [0]*(n//2+3); D[1] = 1
    R = []; b = True; h = 1
    for i in (1..n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k+1]
            R.append(D[1]); h += 1
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1])
        b = not b
    return R
A099960_list(27)  # Peter Luschny, Apr 30 2012

a(n) ~ 2^(5/2) * n^(n+3/2) / (Pi^(n+1/2) * exp(n)). - Vaclav Kotesovec, Sep 10 2014

More terms from Emeric Deutsch, Nov 16 2004

A289312 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n, and each row and column contains at least one nonzero entry.

Original entry on oeis.org

1, 2, 6, 26, 142, 946, 7446, 67658, 697118, 8031586, 102312486, 1427905658, 21666671534, 355138949394, 6253348428598, 117720540700842, 2359368991571518, 50157679523340994, 1127327559500923974, 26709016625807923418, 665292778385210384078

Offset: 0

Peter Bala, Jul 02 2017

A Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and each row and column contains a nonzero entry. See A022493.
Here we consider generalized Fishburn matrices where we allow the Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized Fishburn matrices of size n.
Alternatively, this sequence gives the number of 2-colored Fishburn matrices of size n, that is, ordinary Fishburn matrices of size n where each nonzero entry in the matrix can have one of two different colors.
More generally, if F(x) = Sum_{n >= 0} (Product_{i = 1..n} 1 - 1/(1 + x)^i) is the o.g.f. for primitive Fishburn matrices A138265 (i.e., Fishburn matrices with entries restricted to the set {0,1}) and C(x) := c_1*x + c_2*x^2 + ..., where c_i is a sequence of nonnegative integers, then the composition F(C(x)) is the o.g.f. for colored Fishburn matrices where entry i in the matrix can have one of c_i different colors: c_i = 0 for some i means i does not appear as an entry in the Fishburn matrix. This result is an application of Lemma 2.2.22 of Goulden and Jackson.

			a(2) = 6: The six upper triangular matrices of size 2 with no zero rows or columns are (+-2) and
  /+-1  0\
  |      |.
  \0  +-1/

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.

Alois P. Heinz, Table of n, a(n) for n = 0..300
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.

Cf. A022493, A138265, A289313.

G:= add(mul(1 - ((1-x)/(1+x))^k, k=1..n),n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);
# Peter Bala, Jul 24 2017

m = 21; Sum[Product[1 - ((1-x)/(1+x))^k + O[x]^m, {k, 1, n}], {n, 0, m}] // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2020 *)

G.f.: Sum_{n >= 0} Product_{k = 1..n} 1 - ((1 - x)/(1 + x))^k.
Alternative g.f.: Sum_{n >= 0} ((1 + x)/(1 - x))^(n+1) * Product_{k = 1..n} 1 - ((1 + x)/(1 - x))^k.
G.f.: B(2*x/(1+x)) where B(x) is the g.f. of A022493. - Michael D. Weiner, Feb 28 2019
a(n) ~ 2^(2*n + 5/2) * 3^(n + 3/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - Vaclav Kotesovec, Aug 31 2023

A289313 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.

Original entry on oeis.org

1, 2, 10, 74, 722, 8786, 128218, 2182554, 42456226, 929093538, 22590839466, 604225121258, 17630145814898, 557285515817970, 18970857530674554, 691929648113663802, 26919562120779248962, 1112769248605003393858, 48704349211392743606602

Offset: 0

Peter Bala, Jul 02 2017

A row-Fishburn matrix of size n is defined to be an upper-triangular matrix with nonnegative integer entries which sum to n and such that each row contains a nonzero entry. See A158691.
Here we consider generalized row-Fishburn matrices where we allow the row_Fishburn matrices to have positive and negative nonzero entries. We define the size of a generalized row-Fishburn matrix to be the absolute sum of the matrix entries. This sequence gives the number of generalized row-Fishburn matrices of size n.
Alternatively, this sequence gives the number of 2-colored row-Fishburn matrices of size n, that is, ordinary row-Fishburn matrices of size n where each nonzero entry in the matrix can have one of two different colors.
More generally, if F(x) = Sum_{n >= 0} ( Product_{i = 1..n} (1 + x)^i - 1 ) is the o.g.f. for primitive row-Fishburn matrices A179525 (i.e., row-Fishburn matrices with entries restricted to the set {0,1}) and C(x) := c_1*x + c_2*x^2 + ..., where c_i is a sequence of nonnegative integers, then the composition F(C(x)) is the o.g.f. for colored row-Fishburn matrices where entry i in the matrix can have one of c_i different colors: c_i = 0 for some i means i does not appear as an entry in the Fishburn matrix. This result is an application of Lemma 2.2.22 of Goulden and Jackson.

			a(2) = 10: The ten generalized row-Fishburn matrices of size 2 are
  (+-2),
  /+-1  0\ and  /0 +-1\
  |      |      |     |
  \0  +-1/      \0 +-1/.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 42.

Seiichi Manyama, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A022493, A158691, A179525, A289312.

G:= add(mul( ((1 + x)/(1 - x))^i - 1, i=1..n),n=0..20):
S:= series(G,x,21):
seq(coeff(S,x,j),j=0..20);
# Peter Bala, Jul 24 2017

O.g.f.: Sum_{n >= 0} ( Product_{i = 1..n} ((1 + x)/(1 - x))^i - 1 ).
The o.g.f. has several alternative forms:
Sum_{n >= 0} ( Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^(2*i-1) ) );
Sum_{n >= 0} ((1 - x)/(1 + x))^(n+1) * ( Product_{i = 1..n} 1 - ((1 - x)/(1 + x))^(2*i) );
1/2*( 1 + Sum_{n >= 0} ((1 + x)/(1 - x))^((n+1)*(n+2)/2) * Product_{i = 1..n} ( 1 - ((1 - x)/(1 + x))^i ) ).
Conjectural g.f.: Sum_{n >= 0} ((1 + x)/(1 - x))^((n+1)*(2*n+1)) * Product_{i = 1..2*n} ( ((1 - x)/(1 + x))^i - 1 ).
a(n) ~ 2^(3*n+2) * 3^(n+1) * n^(n + 1/2) / (exp(n) * Pi^(2*n + 3/2)). - Vaclav Kotesovec, Aug 31 2023

A225588 Number of descent sequences of length n.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 67, 222, 832, 3501, 16412, 85062, 484013, 3004342, 20226212, 146930527, 1146389206, 9566847302, 85073695846, 803417121866, 8032911742979, 84796557160893, 942648626858310, 11009672174119829, 134809696481902160, 1727161011322322267, 23110946295566466698, 322435363123261622935

Offset: 0

Joerg Arndt, May 11 2013

nonn

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.
Replacing the function desc(.) by a function chg(.) that counts changes in the prefix gives A000522 (arrangements).
Replacing the function desc(.) by a function asc(.) that counts ascents in the prefix gives A022493 (ascent sequences).
Replacing the function desc(.) by a function eq(.) that counts successive equal parts in the prefix gives A000110 (set partitions).

			The a(5)=23 descent sequences of length 5 are (dots for zeros)
01:  [ . . . . . ]
02:  [ . . . . 1 ]
03:  [ . . . 1 . ]
04:  [ . . . 1 1 ]
05:  [ . . 1 . . ]
06:  [ . . 1 . 1 ]
07:  [ . . 1 . 2 ]
08:  [ . . 1 1 . ]
09:  [ . . 1 1 1 ]
10:  [ . 1 . . . ]
11:  [ . 1 . . 1 ]
12:  [ . 1 . . 2 ]
13:  [ . 1 . 1 . ]
14:  [ . 1 . 1 1 ]
15:  [ . 1 . 1 2 ]
16:  [ . 1 . 2 . ]
17:  [ . 1 . 2 1 ]
18:  [ . 1 . 2 2 ]
19:  [ . 1 1 . . ]
20:  [ . 1 1 . 1 ]
21:  [ . 1 1 . 2 ]
22:  [ . 1 1 1 . ]
23:  [ . 1 1 1 1 ]

Joerg Arndt, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..490 (first 200 terms from Joerg Arndt and Alois P. Heinz)
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.

Cf. A225624 (triangle: descent sequences by numbers of descents).

b:= proc(n, i, t) option remember; `if`(n<1, 1,
      add(b(n-1, j, t+`if`(j b(n-1, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, May 13 2013

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Sum[b[n-1, j, t + If[jJean-François Alcover, Apr 09 2015, after Alois P. Heinz *)

# Program adapted from Alois P. Heinz's Maple code in A022493.
# b(n,i,t) gives the number of length-n postfixes of descent sequences
# with a prefix having t descents and last element i.
@CachedFunction
def b(n,i,t):
    if n<=1: return 1
    return sum( b(n-1, j, t+(j

cp's OEIS Frontend

A202059 Number of ascent sequences avoiding the pattern 100.

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A202060 Number of ascent sequences avoiding the pattern 110.

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A202061 Number of ascent sequences avoiding the pattern 120.

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A218580 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with first occurrence of the maximal value at position k-1.

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A218581 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with last occurrence of the maximal value at position k-1.

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A294220 Number A(n,k) of ascent sequences of length n where no letter multiplicity is larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A099960 An interleaving of the Genocchi numbers of the first and second kind, A110501 and A005439.

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A289312 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n, and each row and column contains at least one nonzero entry.

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A289313 The number of upper-triangular matrices with integer entries whose absolute sum is equal to n and such that each row contains a nonzero entry.

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