A227038 -id:A227038 - cp's OEIS frontend

A072704 Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 7, 5, 1, 1, 6, 12, 12, 9, 6, 1, 1, 7, 16, 20, 16, 11, 7, 1, 1, 8, 21, 30, 28, 20, 13, 8, 1, 1, 9, 27, 42, 45, 36, 24, 15, 9, 1, 1, 10, 33, 58, 68, 60, 44, 28, 17, 10, 1, 1, 11, 40, 77, 98, 95, 75, 52, 32, 19, 11, 1

Offset: 1

Henry Bottomley, Jul 04 2002

			Rows start:
01:  [1]
02:  [1, 1]
03:  [1, 2, 1]
04:  [1, 3, 3, 1]
05:  [1, 4, 5, 4, 1]
06:  [1, 5, 8, 7, 5, 1]
07:  [1, 6, 12, 12, 9, 6, 1]
08:  [1, 7, 16, 20, 16, 11, 7, 1]
09:  [1, 8, 21, 30, 28, 20, 13, 8, 1]
10:  [1, 9, 27, 42, 45, 36, 24, 15, 9, 1]
...
T(6,3)=8 since 6 can be written as 1+1+4, 1+2+3, 1+3+2, 1+4+1, 2+2+2, 2+3+1, 3+2+1, or 4+1+1 but not 2+1+3 or 3+1+2.

Alois P. Heinz, Rows n = 1..141, flattened
Henry Bottomley, Illustration of initial terms
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3094, Table 4.
Eric Weisstein's World of Mathematics, Unimodal Sequence

Cf. A059623, A072705. Row sums are A001523. First column is A057427, second is A000027 offset, third appears to be A000212 offset, right hand columns include A000012, A000027, A005408 and A008574.
The case of partitions is A072233.
Dominates A332670 (the version for negated compositions).
The strict case is A072705.
The case of constant compositions is A113704.
Unimodal sequences covering an initial interval are A007052.
Partitions whose run-lengths are unimodal are A332280.
Cf. A107429, A115981, A227038, A328509, A332282, A332283, A332578, A332638, A332669, A332728.

b:= proc(n, i) option remember; local q; `if`(i>n, 0,
      `if`(irem(n, i, 'q')=0, x^q, 0) +expand(
      add(b(n-i*j, i+1)*(j+1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i ] == 0, x^Quotient[n, i], 0] + Expand[ Sum[b[n-i*j, i+1]*(j+1)*x^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],unimodQ]],{n,0,10},{k,0,n}] (* Gus Wiseman, Mar 06 2020 *)

\\ starting for n=0, with initial column 1, 0, 0, ...:
N=25;  x='x+O('x^N);
T=Vec(1 + sum(n=1, N, t*x^n / ( prod(k=1,n-1, (1 - t*x^k)^2 ) * (1 - t*x^n) ) ) )
for(r=1,#T, print(Vecrev(T[r])) ); \\ Joerg Arndt, Oct 01 2017

G.f. with initial column 1, 0, 0, ...: 1 + Sum_{n>=1} (t*x^n / ( ( Product_{k=1..n-1} (1 - t*x^k)^2 ) * (1 - t*x^n) ) ). - Joerg Arndt, Oct 01 2017

A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 25, 30, 36, 40, 45, 54, 59, 68, 79, 86, 96, 112, 121, 135, 155, 168, 188, 214, 230, 253, 284, 308, 337, 380, 407, 445, 497, 533, 580, 645, 689, 748, 828, 885, 956, 1053, 1124, 1212, 1330, 1415, 1519, 1665, 1771

Offset: 0

Gus Wiseman, Feb 24 2020

nonn

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(1) = 1 through a(9) = 8 partitions:
  1  11  21   211   221    321     2221     3221      3321
         111  1111  2111   2211    3211     22211     22221
                    11111  21111   22111    32111     32211
                           111111  211111   221111    222111
                                   1111111  2111111   321111
                                            11111111  2211111
                                                      21111111
                                                      111111111

MathWorld, Unimodal Sequence

Not requiring unimodality gives A000009.
A version for compositions is A227038.
Not requiring the partition to cover an initial interval gives A332280.
The complement is counted by A332579.
Unimodal compositions are A001523.
Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],normQ[#]&&unimodQ[Length/@Split[#]]&]],{n,0,30}]

A332726 Number of compositions of n whose run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A composition of n is a finite sequence of positive integers summing to n.

			The only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not run-lengths) gives A001523.
The case of partitions is A332280, with complement counted by A332281.
The complement is counted by A332727.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negated run-lengths are unimodal are A332578.
Compositions whose negated run-lengths are not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M,m)); vector(n, i, vecsum(M[i,]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[,1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1,1..n-m+2]; S+=concat(vector(m), desc(D,m))); S} \\ Andrew Howroyd, Dec 31 2020

a(n) + A332727(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A332288 Number of unimodal permutations of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 6, 1, 2, 3, 1, 2, 4, 1, 3, 2, 4, 1, 4, 1, 2, 2, 3, 2, 4, 1, 5, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 22 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also permutations of the multiset of prime indices of n avoiding the patterns (2,1,2), (2,1,3), and (3,1,2).

			The a(n) permutations for n = 2, 6, 12, 24, 48, 60, 120, 180:
  (1)  (12)  (112)  (1112)  (11112)  (1123)  (11123)  (11223)
       (21)  (121)  (1121)  (11121)  (1132)  (11132)  (11232)
             (211)  (1211)  (11211)  (1231)  (11231)  (11322)
                    (2111)  (12111)  (1321)  (11321)  (12231)
                            (21111)  (2311)  (12311)  (12321)
                                     (3211)  (13211)  (13221)
                                             (23111)  (22311)
                                             (32111)  (23211)
                                                      (32211)

Wikipedia, Permutation pattern
MathWorld, Unimodal Sequence

Dominated by A008480.
A more interesting version is A332294.
The complement is counted by A332671.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with unimodal 0-appended first differences are A332283.
Cf. A056239, A112798, A115981, A124010, A227038, A304660, A328509, A332280, A332284, A332294, A332578, A332672.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]],unimodQ]],{n,30}]

A332870 Number of compositions of n that are neither unimodal nor is their negation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 9, 32, 92, 243, 587, 1361, 3027, 6564, 13928, 29127, 60180, 123300, 250945, 508326, 1025977, 2065437, 4150056, 8327344, 16692844, 33438984, 66951671, 134004892, 268148573, 536486146, 1073227893, 2146800237, 4294061970, 8588740071, 17178298617

Offset: 0

Gus Wiseman, Mar 02 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

A composition of n is a finite sequence of positive integers summing to n.

			The a(6) = 2 and a(7) = 9 compositions:
  (1212)  (1213)
  (2121)  (1312)
          (2131)
          (3121)
          (11212)
          (12112)
          (12121)
          (21121)
          (21211)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The case of run-lengths of partitions is A332640.
The version for unsorted prime signature is A332643.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Compositions whose negation is unimodal are A332578.
Compositions whose negation is not unimodal are A332669.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions that are neither weakly increasing nor decreasing are A332834.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Cf. A000005, A000041, A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332641, A332746, A332831, A332833.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[#]&&!unimodQ[-#]&]],{n,0,10}]

a(n) = 2^(n-1) - A001523(n) - A332578(n) + 2*A000041(n) - A000005(n) for n > 0. - Andrew Howroyd, Dec 30 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857

Offset: 0

Gus Wiseman, Feb 21 2020

nonn

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.

			The a(8) = 1 through a(18) = 7 partitions:
  (431)  .  (541)  (641)  (651)   (652)   (752)   (762)   (862)
                          (5421)  (751)   (761)   (861)   (871)
                                  (5431)  (851)   (6531)  (961)
                                          (6431)  (7431)  (6532)
                                          (6521)  (7521)  (6541)
                                                          (7621)
                                                          (8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Strict partitions are A000009.
Partitions covering an initial interval are (also) A000009.
The non-strict version is A332284.
The complement is counted by A332285.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Normal partitions whose run-lengths are not unimodal are A332579.
Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!unimodQ[Differences[Append[#,0]]]]&]],{n,0,30}]

A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 5, 2, 1, 0, 1, 5, 7, 5, 2, 1, 0, 1, 6, 11, 10, 5, 2, 1, 0, 1, 7, 15, 16, 10, 5, 2, 1, 0, 1, 8, 20, 24, 20, 10, 5, 2, 1, 0, 1, 9, 25, 36, 31, 20, 10, 5, 2, 1, 0, 1, 10, 32, 50, 50, 36, 20, 10, 5, 2, 1

Offset: 0

Gus Wiseman, Feb 29 2020

A composition of n is a finite sequence of positive integers summing to n.

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  2  1
  0  1  3  2  1
  0  1  4  5  2  1
  0  1  5  7  5  2  1
  0  1  6 11 10  5  2  1
  0  1  7 15 16 10  5  2  1
  0  1  8 20 24 20 10  5  2  1
  0  1  9 25 36 31 20 10  5  2  1
  0  1 10 32 50 50 36 20 10  5  2  1
  0  1 11 38 67 73 59 36 20 10  5  2  1
Column n = 7 counts the following compositions:
  (7)  (16)  (115)  (1114)  (11113)  (111112)  (1111111)
       (25)  (124)  (1123)  (11122)  (211111)
       (34)  (133)  (1222)  (21112)
       (43)  (214)  (2113)  (22111)
       (52)  (223)  (2122)  (31111)
       (61)  (313)  (2212)
             (322)  (2221)
             (331)  (3112)
             (412)  (3211)
             (421)  (4111)
             (511)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The case of partitions is A072233.
Dominated by A072704 (the non-negated version).
The strict case is A072705.
The case of constant compositions is A113704.
Row sums are A332578.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose negated unsorted prime signature is not unimodal are A332282.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions whose negation is not unimodal are A332669.
Partitions whose negated 0-appended first differences are unimodal: A332728.
Cf. A011782, A107429, A227038, A332280, A332283, A332639, A332642, A332741, A332742, A332744, A332832, A332870.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],unimodQ[-#]&]],{n,0,10},{k,0,n}]

T(n)={[Vecrev(p) | p<-Vec(1 + sum(j=1, n, y*x^j/((1-y*x^j) * prod(k=j+1, n-j, 1 - y*x^k + O(x*x^(n-j)))^2)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

G.f.: A(x,y) = 1 + Sum_{j>0} y*x^j/((1 - y*x^j)*Product_{k>j} (1 - y*x^k)^2). - Andrew Howroyd, Jan 11 2024

A332285 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 8, 9, 11, 13, 15, 17, 22, 25, 29, 34, 39, 42, 53, 58, 64, 75, 84, 93, 111, 122, 134, 152, 169, 184, 212, 232, 252, 287, 315, 342, 389, 419, 458, 512, 556, 602, 672, 727, 787, 870, 940, 1012, 1124, 1209, 1303, 1431, 1540, 1655, 1821

Offset: 0

Gus Wiseman, Feb 21 2020

nonn

First differs from A000009 at a(8) = 5, A000009(8) = 6.

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

			The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)
            (21)  (31)  (32)  (42)   (43)   (53)   (54)
                        (41)  (51)   (52)   (62)   (63)
                              (321)  (61)   (71)   (72)
                                     (421)  (521)  (81)
                                                   (432)
                                                   (531)
                                                   (621)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is not counted under a(8).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The non-strict version is A332283.
The complement is counted by A332286.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Unimodal permutations are A011782.
Partitions with unimodal run-lengths are A332280.
Cf. A025065, A072706, A115981, A227038, A332282, A332284, A332287, A332288, A332577, A332638, A332642.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,unimodQ[Differences[Append[#,0]]]]&]],{n,0,30}]

A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 7, 8, 10, 14, 19, 22, 30, 36, 43, 56, 69, 80, 101, 121, 141, 172, 202, 234, 282, 332, 384, 452, 527, 602, 706, 815, 929, 1077, 1236, 1403, 1615, 1842, 2082, 2379, 2702, 3044, 3458, 3908, 4388, 4963, 5589, 6252

Offset: 0

Gus Wiseman, Feb 25 2020

nonn

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Also the number of strict integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

			The a(10) = 1 through a(16) = 7 partitions:
  33211  332111  3321111  333211    433211     443211      443221
                          33211111  3332111    4332111     3333211
                                    332111111  33321111    4432111
                                               3321111111  33322111
                                                           43321111
                                                           333211111
                                                           33211111111

MathWorld, Unimodal Sequence

The complement is counted by A332577.
Not requiring the partition to cover an initial interval gives A332281.
The opposite version is A332286.
A version for compositions is A332743.
Partitions covering an initial interval of positive integers are A000009.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negated run-lengths are not unimodal are A332727.
Cf. A007052, A100883, A107429, A227038, A332280, A332284, A332638, A332639, A332640, A332671, A332672, A332728.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&!unimodQ[Length/@Split[#]]&]],{n,0,30}]

A056242 Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 16, 8, 1, 14, 41, 44, 16, 1, 20, 85, 146, 112, 32, 1, 27, 155, 377, 456, 272, 64, 1, 35, 259, 833, 1408, 1312, 640, 128, 1, 44, 406, 1652, 3649, 4712, 3568, 1472, 256, 1, 54, 606, 3024, 8361, 14002, 14608, 9312, 3328, 512, 1, 65, 870, 5202

Offset: 1

Colin Mallows, Aug 23 2000

Generalized Riordan array (1/(1-x), x/(1-x) + x*dif(x/1-x),x)). - Paul Barry, Dec 26 2007
Reversal of A117317. - Philippe Deléham, Feb 11 2012
Essentially given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 11 2012
This sequence is given in the Strehl presentation with the o.g.f. (1-z)/[1-2(1+t)z+(1+t)z^2], with offset 0, along with a recursion relation, a combinatorial interpretation, and relations to Hermite and Laguerre polynomials. Note that the o.g.f. is related to that of A049310. - Tom Copeland, Jan 08 2017
From Gus Wiseman, Mar 06 2020: (Start)
T(n,k) is also the number of unimodal length-n sequences covering an initial interval of positive integers with maximum part k, where a sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. For example, the sequences counted by row n = 4 are:
  (1111)  (1112)  (1123)  (1234)
          (1121)  (1132)  (1243)
          (1122)  (1223)  (1342)
          (1211)  (1231)  (1432)
          (1221)  (1232)  (2341)
          (1222)  (1233)  (2431)
          (2111)  (1321)  (3421)
          (2211)  (1322)  (4321)
          (2221)  (1332)
                  (2231)
                  (2311)
                  (2321)
                  (2331)
                  (3211)
                  (3221)
                  (3321)
(End)
T(n,k) is the number of hexagonal directed-column convex polyominoes of area n with k columns (see Baril et al. at page 9). - Stefano Spezia, Oct 14 2023

			Triangle begins:
  1;
  1,    2;
  1,    5,    4;
  1,    9,   16,    8;
  1,   14,   41,   44,   16;
  1,   20,   85,  146,  112,   32;
  1,   27,  155,  377,  456,  272,   64;
  1,   35,  259,  833, 1408, 1312,  640,  128;
  1,   44,  406, 1652, 3649, 4712, 3568, 1472,  256;
T(3,2)=5 because we have {1}{23}, {23}{1}, {12}{3}, {3}{12} and {2}{13}.
Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   5,   4,   0;
  1,   9,  16,   8,   0;
  1,  14,  41,  44,  16,   0;
  1,  20,  85, 146, 112,  32,   0;
  1,  27, 155, 377, 456, 272,  64,   0;

Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
Finn Bjarne Jost, Tautological Intersection Numbers and Order-Consecutive Partition Sequences, arXiv:2307.15825 [math.CO], 2023. See p. 9.
V. Strehl, Combinatoire rétrospective et créative, an on-line presentation, slide 36, SLC 71, Bertinoro,, September 18, 2013.
Volker Strehl, Lacunary Laguerre Series from a Combinatorial Perspective, Séminaire Lotharingien de Combinatoire, B76c (2017).

Row sums are A007052.
Column k = n - 1 is A053220.
Ordered set-partitions are A000670.
Cf. A001523, A049310, A072704, A084938, A097805, A117317, A227038, A328509, A332294, A332673, A332724, A332872.

a056242 n k = a056242_tabl !! (n-1)!! (k-1)
a056242_row n = a056242_tabl !! (n-1)
a056242_tabl = [1] : [1,2] : f [1] [1,2] where
   f us vs = ws : f vs ws where
     ws = zipWith (-) (map (* 2) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))
                      (zipWith (+) ([0] ++ us ++ [0]) (us ++ [0,0]))
-- Reinhard Zumkeller, May 08 2014

T:=proc(n,k) if k=1 then 1 elif k<=n then sum((-1)^(k-1-j)*binomial(k-1,j)*binomial(n+2*j-1,2*j),j=0..k-1) else 0 fi end: seq(seq(T(n,k),k=1..n),n=1..12);

rows = 11; t[n_, k_] := (-1)^(k+1)*HypergeometricPFQ[{1-k, (n+1)/2, n/2}, {1/2, 1}, 1]; Flatten[ Table[ t[n, k], {n, 1, rows}, {k, 1, n}]](* Jean-François Alcover, Nov 17 2011 *)

The Hwang and Mallows reference gives explicit formulas.
T(n,k) = Sum_{j=0..k-1} (-1)^(k-1-j)*binomial(k-1, j)*binomial(n+2j-1, 2j) (1<=k<=n); this is formula (11) in the Huang and Mallows reference.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(1,1) = 1, T(2,1) = 1, T(2,2) = 2. - Philippe Deléham, Feb 11 2012
G.f.: -(-1+x)*x*y/(1-2*x-2*x*y+x^2*y+x^2). - R. J. Mathar, Aug 11 2015

cp's OEIS Frontend

A072704 Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms.

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A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

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A332726 Number of compositions of n whose run-lengths are unimodal.

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A332288 Number of unimodal permutations of the multiset of prime indices of n.

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A332870 Number of compositions of n that are neither unimodal nor is their negation.

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A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

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A332670 Triangle read by rows where T(n,k) is the number of length-k compositions of n whose negation is unimodal.

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