A181796 -id:A181796 - cp's OEIS frontend

A335516 Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 7, 2, 3, 3, 7, 2, 4, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3

Offset: 1

Gus Wiseman, Jun 26 2020

nonn

First differs from A181796 at a(180) = 9, A181796(180) = 10.
First differs from A335549 at a(90) = 7, A335549(90) = 8.
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.
We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

			The a(n) patterns for n = 2, 30, 12, 60, 120, 540, 1500:
  ()   ()     ()     ()      ()       ()        ()
  (1)  (1)    (1)    (1)     (1)      (1)       (1)
       (12)   (11)   (11)    (11)     (11)      (11)
       (123)  (12)   (12)    (12)     (12)      (12)
              (112)  (112)   (111)    (111)     (111)
                     (123)   (112)    (112)     (112)
                     (1123)  (123)    (122)     (122)
                             (1112)   (1112)    (123)
                             (1123)   (1122)    (1123)
                             (11123)  (1222)    (1222)
                                      (11222)   (1233)
                                      (12223)   (11233)
                                      (112223)  (12333)
                                                (112333)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for standard compositions is A335458.
The not necessarily contiguous version is A335549.
Patterns are counted by A000670 and ranked by A333217.
A number's prime indices are given in the rows of A112798.
Contiguous subsequences of standard compositions are A124771.
Contiguous sub-partitions of prime indices are counted by A335519.
Minimal avoided patterns of prime indices are counted by A335550.
Patterns contiguously matched by partitions are counted by A335838.
Cf. A000005, A056239, A056986, A108917, A124770, A181796, A269134, A333224, A334299, A335457, A335837.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[primeMS[n],{_,s___,_}:>{s}]]],{n,100}]

A336415 Number of divisors of n! with equal prime multiplicities.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 13, 21, 24, 28, 33, 49, 53, 85, 94, 100, 104, 168, 173, 301, 307, 317, 334, 590, 595, 603, 636, 642, 652, 1164, 1171, 2195, 2200, 2218, 2283, 2295, 2301, 4349, 4478, 4512, 4519, 8615, 8626, 16818, 16836, 16844, 17101, 33485, 33491, 33507, 33516, 33582

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number k has "equal prime multiplicities" (or is "uniform") iff its prime signature is constant, meaning that k is a power of a squarefree number.

			The a(n) uniform divisors of n for n = 1, 2, 6, 8, 30, 36 are the columns:
  1  2  6  8  30  36
     1  3  6  15  30
        2  4  10  16
        1  3   8  15
           2   6  10
           1   5   9
               4   8
               3   6
               2   5
               1   4
                   3
                   2
                   1
In 20!, the multiplicity of the third prime (5) is 4 but the multiplicity of the fourth prime (7) is 2. Hence there are 2^3 - 1 = 3 divisors with all exponents 3 (we subtract |{1}| = 1 from that count as 1 has no exponent 3). - _David A. Corneth_, Jul 27 2020

David A. Corneth, Table of n, a(n) for n = 0..9999
Jon Maiga, Computer-generated formulas for A336415, Sequence Machine.

The version for distinct prime multiplicities is A336414.
The version for nonprime perfect powers is A336416.
Uniform partitions are counted by A047966.
Uniform numbers are A072774, with nonprime terms A182853.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
Maximum divisor with distinct prime multiplicities is A327498.
Uniform divisors are counted by A327527.
Maximum uniform divisor is A336618.
1st differences are given by A048675.
Cf. A000005, A000961, A001222, A001597, A007916, A112798, A124010, A327499.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617.

Table[Length[Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); (#ex==0) || (vecmin(ex) == vecmax(ex))); \\ Michel Marcus, Jul 24 2020

a(n) = {if(n<2, return(1)); my(f = primes(primepi(n)), res = 1, t = #f); f = vector(#f, i, val(n, f[i])); for(i = 1, f[1], while(f[t] < i, t--; ); res+=(1<David A. Corneth, Jul 27 2020

a(n) = A327527(n!).

Terms a(31) and onwards from David A. Corneth, Jul 27 2020

A335446 Number of (1,2,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 13 2020

nonn

Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 120, 144:
  (121)  (1121)  (1212)  (1213)  (11212)  (1232)  (11213)  (111212)
         (1211)  (1221)  (1231)  (11221)  (2132)  (11231)  (111221)
                 (2121)  (1312)  (12112)  (2312)  (11312)  (112112)
                         (1321)  (12121)  (2321)  (11321)  (112121)
                         (2131)  (12211)          (12113)  (112211)
                         (3121)  (21121)          (12131)  (121112)
                                 (21211)          (12311)  (121121)
                                                  (13112)  (121211)
                                                  (13121)  (122111)
                                                  (13211)  (211121)
                                                  (21131)  (211211)
                                                  (21311)  (212111)
                                                  (31121)
                                                  (31211)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of zeros are A065200.
The avoiding version is A335449.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
Unimodal permutations of prime indices are counted by A332288.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(1,2,1)-matching compositions are ranked by A335466.
(1,2,1)-matching compositions are counted by A335470.
(1,2,1)-matching patterns are counted by A335509.
Cf. A056239, A056986, A112798, A158005, A158009, A181796, A335452, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x

A336419 Number of divisors d of the n-th superprimorial A006939(n) with distinct prime exponents such that the quotient A006939(n)/d also has distinct prime exponents.

Original entry on oeis.org

1, 2, 4, 10, 24, 64, 184, 536, 1608, 5104, 16448, 55136, 187136, 658624, 2339648, 8618208, 31884640, 121733120, 468209408, 1849540416, 7342849216

Offset: 0

Gus Wiseman, Jul 25 2020

A number has distinct prime exponents iff its prime signature is strict.

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(0) = 1 through a(3) = 10 divisors:
  1  2  12  360
-----------------
  1  1   1    1
     2   3    5
         4    8
        12    9
             18
             20
             40
             45
             72
            360

A000110 shifted once to the left dominates this sequence.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
Cf. A000005, A000178, A008278, A071625, A076954, A118914, A124010, A327498, A327527, A336420, A336421, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
Table[Length[Select[Divisors[chern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,6}]

recurse(n,k,b,d)={if(k>n, 1, sum(i=0, k, if((i==0||!bittest(b,i)) && (i==k||!bittest(d,k-i)), self()(n, k+1, bitor(b, 1<Andrew Howroyd, Aug 30 2020

a(10)-a(20) from Andrew Howroyd, Aug 31 2020

A048742 a(n) = n! - (n-th Bell number).

Original entry on oeis.org

0, 0, 0, 1, 9, 68, 517, 4163, 36180, 341733, 3512825, 39238230, 474788003, 6199376363, 86987391878, 1306291409455, 20912309745853, 355604563226196, 6401691628921841, 121639267666626943, 2432850284018404628, 51090467301893283249, 1123996221061869232677

Offset: 0

N. J. A. Sloane

nonn

Number of permutations of [n] which have at least one cycle that has at least one inversion when written with its smallest element in the first position. Example: a(4)=9 because we have (1)(243), (1432), (142)(3), (132)(4), (1342), (1423), (1243), (143)(2) and (1324). - Emeric Deutsch, Apr 29 2008
Number of permutations of [n] having consecutive runs of increasing elements with initial elements in increasing order. a(4) = 9: `124`3, `13`24, `134`2, `14`23, `14`3`2, `2`14`3, `24`3`1, `3`14`2, `4`13`2. - Alois P. Heinz, Apr 27 2016
From Gus Wiseman, Aug 11 2020: (Start)
Also the number of divisors of the superfactorial A006939(n - 1) without distinct prime multiplicities. For example, the a(4) = 9 divisors together with their prime signatures are the following. Note that A076954 can be used here instead of A006939.
    6: (1,1)
   10: (1,1)
   15: (1,1)
   30: (1,1,1)
   36: (2,2)
   60: (2,1,1)
   90: (1,2,1)
  120: (3,1,1)
  180: (2,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..450
Index entries for sequences related to factorial numbers

A000110 lists Bell numbers.
A000142 lists factorial numbers.
A006939 lists superprimorials or Chernoff numbers.
A181796 counts divisors with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
Cf. A000005, A022915, A027423, A076954, A095149, A130091, A336420, A336421.

with(combinat): seq(factorial(n)-bell(n),n=0..21); # Emeric Deutsch, Apr 29 2008

Table[n! - BellB[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)

[factorial(m) - bell_number(m) for m in range(23)]  # Zerinvary Lajos, Jul 06 2008

a(n) = A000142(n) - A000110(n).

E.g.f.: 1/(1-x) - exp(exp(x)-1). - Alois P. Heinz, Apr 27 2016

A336422 Number of ways to choose a divisor of a divisor of n, both having distinct prime exponents.

Original entry on oeis.org

1, 3, 3, 6, 3, 5, 3, 10, 6, 5, 3, 13, 3, 5, 5, 15, 3, 13, 3, 13, 5, 5, 3, 24, 6, 5, 10, 13, 3, 7, 3, 21, 5, 5, 5, 21, 3, 5, 5, 24, 3, 7, 3, 13, 13, 5, 3, 38, 6, 13, 5, 13, 3, 24, 5, 24, 5, 5, 3, 20, 3, 5, 13, 28, 5, 7, 3, 13, 5, 7, 3, 42, 3, 5, 13, 13, 5, 7, 3

Offset: 1

Gus Wiseman, Jul 26 2020

nonn

A number has distinct prime exponents iff its prime signature is strict.

			The a(n) ways for n = 1, 2, 4, 6, 8, 12, 30, 210:
  1/1/1  2/1/1  4/1/1  6/1/1  8/1/1  12/1/1    30/1/1  210/1/1
         2/2/1  4/2/1  6/2/1  8/2/1  12/2/1    30/2/1  210/2/1
         2/2/2  4/2/2  6/2/2  8/2/2  12/2/2    30/2/2  210/2/2
                4/4/1  6/3/1  8/4/1  12/3/1    30/3/1  210/3/1
                4/4/2  6/3/3  8/4/2  12/3/3    30/3/3  210/3/3
                4/4/4         8/4/4  12/4/1    30/5/1  210/5/1
                              8/8/1  12/4/2    30/5/5  210/5/5
                              8/8/2  12/4/4            210/7/1
                              8/8/4  12/12/1           210/7/7
                              8/8/8  12/12/2
                                     12/12/3
                                     12/12/4
                                     12/12/12

A336421 is the case of superprimorials.
A007425 counts divisors of divisors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor with distinct prime exponents.
A336500 counts divisors with quotient also having distinct prime exponents.
A336568 = not a product of two numbers with distinct prime exponents.
Cf. A000005, A001055, A005117, A032741, A071625, A118914, A124010, A336419, A336420.

strdivs[n_]:=Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&];
Table[Sum[Length[strdivs[d]],{d,strdivs[n]}],{n,30}]

A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 1, 2, 2, 0, 1, 4, 1, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) chains for n = 12, 72, 144, 192 (ones not shown):
  12/3    72/18/2       144/72/18/2       192/96/48/24/12/3
  12/4/2  72/18/9/3     144/72/18/9/3     192/64/32/16/8/4/2
          72/24/12/3    144/48/24/12/3    192/96/32/16/8/4/2
          72/24/8/4/2   144/72/24/12/3    192/96/48/16/8/4/2
          72/24/12/4/2  144/48/16/8/4/2   192/96/48/24/8/4/2
                        144/48/24/8/4/2   192/96/48/24/12/4/2
                        144/72/24/8/4/2
                        144/48/24/12/4/2
                        144/72/24/12/4/2

A336423 is the non-maximal version.
A336570 is the version for chains not necessarily containing n.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336571 counts divisor sets of elements of A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strchs[n_]:=If[n==1,{{}},If[!strsigQ[n],{},Join@@Table[Prepend[#,d]&/@strchs[d],{d,Select[Most[Divisors[n]],strsigQ]}]]];
Table[Length[fasmax[strchs[n]]],{n,100}]

A335549 Number of normal patterns matched by the multiset of prime indices of n in weakly increasing order.

Original entry on oeis.org

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

First differs from A181796 at a(90) = 8 A181796(90) = 7.
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.
We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The Heinz number of (1,2,2,3) is 90 and it matches 8 patterns: (), (1), (11), (12), (112), (122), (123), (1223); so a(90) = 8.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for standard compositions instead of prime indices is A335454.
Permutations of prime indices are counted by A008480.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Subset-sums are counted by A304792 and ranked by A299701.
Patterns matched by compositions of n are counted by A335456(n).
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A056239, A108917, A112798, A333221, A335452, A335457, A335458.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@Subsets[primeMS[n]]]],{n,100}]

A095149 Triangle read by rows: Aitken's array (A011971) but with a leading diagonal before it given by the Bell numbers (A000110), 1, 1, 2, 5, 15, 52, ...

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 7, 10, 15, 52, 15, 20, 27, 37, 52, 203, 52, 67, 87, 114, 151, 203, 877, 203, 255, 322, 409, 523, 674, 877, 4140, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 21147, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset: 0

Gary W. Adamson, May 30 2004

Or, prefix Aitken's array (A011971) with a leading diagonal of 0's and take the differences of each row to get the new triangle.
With offset 1, triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} in which k is the largest entry in the block containing 1 (1 <= k <= n). - Emeric Deutsch, Oct 29 2006
Row term sums = the Bell numbers starting with A000110(1): 1, 2, 5, 15, ...
The k-th term in the n-th row is the number of permutations of length n starting with k and avoiding the dashed pattern 23-1. Equivalently, the number of permutations of length n ending with k and avoiding 1-32. - Andrew Baxter, Jun 13 2011
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: Also the number of divisors d with distinct prime multiplicities of the superprimorial A006939(n) that are of the form d = m * 2^k where m is odd. For example, row n = 4 counts the following divisors:
  1     2     4    8     16
  3     18    12   24    48
  5     50    20   40    80
  7     54    28   56    112
  9     1350  108  72    144
  25          540  200   400
  27          756  360   432
  45               504   720
  63               600   1008
  75               1400  1200
  135                    2160
  175                    2800
  189                    3024
  675                    10800
  4725                   75600
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i whose nonzero values are distinct and such that v_n = k.
Crossrefs:
A008278 is the version counted by omega A001221.
A336420 is the version counted by Omega A001222.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by Omega.
A022915 counts permutations of prime indices of superprimorials.
A098859 counts partitions with distinct multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
Cf. A000005, A000142, A027423, A076954, A124010, A146291, A181818, A336417, A336419, A336421, A336499, A336942.
(End)

			Triangle starts:
   1;
   1,  1;
   2,  1,  2;
   5,  2,  3,  5;
  15,  5,  7, 10, 15;
  52, 15, 20, 27, 37, 52;
From _Gus Wiseman_, Aug 11 2020: (Start)
Row n = 3 counts the following set partitions (described in Emeric Deutsch's comment above):
  {1}{234}      {12}{34}    {123}{4}    {1234}
  {1}{2}{34}    {12}{3}{4}  {13}{24}    {124}{3}
  {1}{23}{4}                {13}{2}{4}  {134}{2}
  {1}{24}{3}                            {14}{23}
  {1}{2}{3}{4}                          {14}{2}{3}
(End)

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from Chai Wah Wu)
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Anders Claesson, Generalized pattern avoidance, Europ. J. Combin., 22 7 (2001), 961-971. See Proposition 3.
A. Bernini, M. Bouvel and L. Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18 (2007), No. 3-4, pp. 223-237 (see transposed array p. 227).

Cf. A000110, A005493, A008278, A011971, A188919, A271466.

T(2n,n) gives A020556.

with(combinat): T:=proc(n,k) if k=1 then bell(n-1) elif k=2 and n>=2 then bell(n-2) elif k<=n then add(binomial(k-2,i)*bell(n-2-i),i=0..k-2) else 0 fi end: matrix(8,8,T): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
Q[1]:=t*s: for n from 2 to 11 do Q[n]:=expand(t^n*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)-Q[n-1]+s*Q[n-1]) od: for n from 1 to 11 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Oct 29 2006
A011971 := proc(n,k) option remember ; if k = 0 then if n=0 then 1; else A011971(n-1,n-1) ; fi ; else A011971(n,k-1)+A011971(n-1,k-1) ; fi ; end: A000110 := proc(n) option remember; if n<=1 then 1 ; else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1) ; fi ; end: A095149 := proc(n,k) option remember ; if k = 0 then A000110(n) ; else A011971(n-1,k-1) ; fi ; end: for n from 0 to 11 do for k from 0 to n do printf("%d, ",A095149(n,k)) ; od ; od ; # R. J. Mathar, Feb 05 2007
# alternative Maple program:
b:= proc(n, m, k) option remember; `if`(n=0, 1, add(
      b(n-1, max(j, m), max(k-1, -1)), j=`if`(k=0, m+1, 1..m+1)))
    end:
T:= (n, k)-> b(n, 0, n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 20 2018

nmax = 10; t[n_, 1] = t[n_, n_] = BellB[n-1]; t[n_, 2] = BellB[n-2]; t[n_, k_] /; n >= k >= 3 := t[n, k] = t[n, k-1] + t[n-1, k-1]; Flatten[ Table[ t[n, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011, from formula with offset 1 *)

# requires Python 3.2 or higher.
from itertools import accumulate
A095149_list, blist = [1,1,1], [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A095149_list += [blist[-1]]+ blist
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

With offset 1, T(n,1) = T(n,n) = T(n+1,2) = B(n-1) = A000110(n-1) (the Bell numbers). T(n,k) = T(n,k-1) + T(n-1,k-1) for n >= k >= 3. T(n,n-1) = B(n-1) - B(n-2) = A005493(n-3) for n >= 3 (B(q) are the Bell numbers A000110). T(n,k) = A011971(n-2,k-2) for n >= k >= 2. In other words, deleting the first row and first column we obtain Aitken's array A011971 (also called Bell or Pierce triangle; offset in A011971 is 0). - Emeric Deutsch, Oct 29 2006

T(n,1) = B(n-1); T(n,2) = B(n-2) for n >= 2; T(n,k) = Sum_{i=0..k-2} binomial(k-2,i)*B(n-2-i) for 3 <= k <= n, where B(q) are the Bell numbers (A000110). Generating polynomial of row n is P[n](t) = Q[n](t,1), where Q[n](t,s) = t^n*Q[n-1](1,s) + s*dQ[n-1](t,s)/ds + (s-1) Q[n-1](t,s); Q[1](t,s) = ts. - Emeric Deutsch, Oct 29 2006

Corrected and extended by R. J. Mathar, Feb 05 2007

Entry revised by N. J. A. Sloane, Jun 01 2005, Jun 16 2007

A335455 Number of compositions of n with some part appearing more than twice.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 11, 30, 69, 142, 334, 740, 1526, 3273, 6840, 14251, 29029, 59729, 122009, 248070, 500649, 1012570, 2040238, 4107008, 8257466, 16562283, 33229788, 66621205, 133478437, 267326999, 535146239, 1071183438, 2143604313, 4289194948, 8581463248

Offset: 0

Gus Wiseman, Jun 15 2020

nonn

Also the number of compositions of n matching the pattern (1,1,1).

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

			The a(3) = 1 through a(6) = 11 compositions:
  (111)  (1111)  (1112)   (222)
                 (1121)   (1113)
                 (1211)   (1131)
                 (2111)   (1311)
                 (11111)  (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The case of partitions is A000726.
The avoiding version is A232432.
The (1,1)-matching version is A261982.
The version for patterns is A335508.
The version for prime indices is A335510.
These compositions are ranked by A335512.
Compositions are counted by A011782.
Combinatory separations are counted by A269134.
Normal patterns matched by compositions are counted by A335456.
Cf. A000740, A034691, A056986, A080599, A106356, A181796, A232464, A238279, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Max@@Length/@Split[Sort[#]]>=3&]],{n,0,10}]

a(n > 0) = 2^(n - 1) - A232432(n).

cp's OEIS Frontend

A335516 Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

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A336415 Number of divisors of n! with equal prime multiplicities.

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A335446 Number of (1,2,1)-matching permutations of the prime indices of n.

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A336419 Number of divisors d of the n-th superprimorial A006939(n) with distinct prime exponents such that the quotient A006939(n)/d also has distinct prime exponents.

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A048742 a(n) = n! - (n-th Bell number).

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A336422 Number of ways to choose a divisor of a divisor of n, both having distinct prime exponents.

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A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

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A335549 Number of normal patterns matched by the multiset of prime indices of n in weakly increasing order.

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