A098504 -id:A098504 - cp's OEIS frontend

A329741 Number of compositions of n whose multiplicities cover an initial interval of positive integers.

1, 1, 1, 3, 6, 11, 14, 34, 52, 114, 225, 464, 539, 1183, 1963, 3753, 6120, 11207, 19808, 38254, 77194, 147906, 224853, 374216, 611081, 1099933, 2129347, 3336099, 5816094, 9797957, 17577710, 29766586, 53276392, 93139668, 163600815, 324464546, 637029845, 1010826499

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(6) = 14 compositions:
  (1)  (2)  (3)    (4)      (5)      (6)
            (1,2)  (1,3)    (1,4)    (1,5)
            (2,1)  (3,1)    (2,3)    (2,4)
                   (1,1,2)  (3,2)    (4,2)
                   (1,2,1)  (4,1)    (5,1)
                   (2,1,1)  (1,1,3)  (1,1,4)
                            (1,2,2)  (1,2,3)
                            (1,3,1)  (1,3,2)
                            (2,1,2)  (1,4,1)
                            (2,2,1)  (2,1,3)
                            (3,1,1)  (2,3,1)
                                     (3,1,2)
                                     (3,2,1)
                                     (4,1,1)

Looking at run-lengths instead of multiplicities gives A329766.
The complete case is A329748.
Complete compositions are A107429.
Cf. A000740, A008965, A098504, A242882, A244164, A329738, A329739, A329740.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[Length/@Split[Sort[#]]]&]],{n,20}]

a(0), a(21)-a(37) from Alois P. Heinz, Nov 21 2019

A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 6, 6, 1, 1, 0, 4, 9, 15, 3, 1, 0, 2, 16, 22, 22, 1, 1, 0, 0, 8, 37, 38, 41, 3, 1, 0, 0, 0, 26, 86, 69, 72, 2, 1, 0, 0, 0, 2, 78, 175, 124, 129, 3, 1, 0, 0, 0, 0, 14, 202, 367, 226, 213, 1, 1, 0, 0, 0, 0, 0, 52, 469, 750, 376, 395, 5, 1

Offset: 1

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   2   1   1
   2   3   2   1
   2   6   6   1   1
   0   4   9  15   3   1
   0   2  16  22  22   1   1
   0   0   8  37  38  41   3   1
   0   0   0  26  86  69  72   2   1
   0   0   0   2  78 175 124 129   3   1
   0   0   0   0  14 202 367 226 213   1   1
   0   0   0   0   0  52 469 750 376 395   5   1
Row n = 6 counts the following compositions:
  (1,1,3,1)    (1,1,4)      (1,5)      (3,3)          (6)
  (1,3,1,1)    (4,1,1)      (2,4)      (2,2,2)
  (1,1,1,2,1)  (1,1,1,3)    (4,2)      (1,1,1,1,1,1)
  (1,2,1,1,1)  (1,2,2,1)    (5,1)
               (2,1,1,2)    (1,2,3)
               (3,1,1,1)    (1,3,2)
               (1,1,1,1,2)  (1,4,1)
               (1,1,2,1,1)  (2,1,3)
               (2,1,1,1,1)  (2,3,1)
                            (3,1,2)
                            (3,2,1)
                            (1,1,2,2)
                            (1,2,1,2)
                            (2,1,2,1)
                            (2,2,1,1)

Row sums are A000079.
Column sums are A329768.
The version with rows reversed is A329744.
Cf. A000740, A008965, A098504, A242882, A318928, A329745, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-k&]],{n,10},{k,n}]

A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369

Offset: 0

Gus Wiseman, Nov 23 2019

A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(5) = 2 through a(8) = 17 compositions:
  (1112)  (1113)   (1114)    (1115)
  (2111)  (1122)   (1222)    (1133)
          (2211)   (2221)    (3311)
          (3111)   (4111)    (5111)
          (11211)  (11122)   (11222)
                   (11311)   (11411)
                   (21112)   (12221)
                   (22111)   (21113)
                   (111121)  (22211)
                   (121111)  (31112)
                             (111131)
                             (111221)
                             (112112)
                             (112211)
                             (122111)
                             (131111)
                             (211211)
For example, the runs-resistance of (111221) is 3 because we have: (111221) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have: (111221) -> (112) -> (1) -> (), so (111221) is counted under a(8).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

The version for binary expansion is A329865.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==degdep[#]&]],{n,0,10}]

A333629 Least k such that the runs-resistance of the k-th composition in standard order is n.

Original entry on oeis.org

1, 3, 5, 11, 27, 93, 859, 13789, 1530805, 1567323995

Offset: 0

Gus Wiseman, Mar 31 2020

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			The sequence together with the corresponding compositions begins:
        1: (1)
        3: (1,1)
        5: (2,1)
       11: (2,1,1)
       27: (1,2,1,1)
       93: (2,1,1,2,1)
      859: (1,2,2,1,2,1,1)
    13789: (1,2,2,1,1,2,1,1,2,1)
  1530805: (2,1,1,2,2,1,2,1,1,2,1,2,2,1)
For example, starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of first appearances in A333628 = number of times applying A333627 to reach a power of 2, starting with n.
A subsequence of A333630.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

nn=1000;
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcrun[n_]:=Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2;
seq=Table[Length[NestWhileList[stcrun,n,Length[stc[#]]>1&]]-1,{n,nn}];
Table[Position[seq,i][[1,1]],{i,Union[seq]}]

a(9) from Amiram Eldar, Aug 04 2025

A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 2, 3, 3, 6, 12, 12, 42, 114, 210, 60, 360, 720, 1320, 1590, 3690, 6450, 16110, 33120, 59940, 61320, 112980, 171780, 387240, 803880, 769440, 1773240, 2823240, 5790960, 9916200, 19502280, 28244160, 56881440, 130548600, 279578880, 320554080, 541323720

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.

			The a(1) = 1 through a(8) = 12 compositions (empty column not shown):
  (1)  (12)  (112)  (122)  (123)  (1123)  (1223)
       (21)  (121)  (212)  (132)  (1132)  (1232)
             (211)  (221)  (213)  (1213)  (1322)
                           (231)  (1231)  (2123)
                           (312)  (1312)  (2132)
                           (321)  (1321)  (2213)
                                  (2113)  (2231)
                                  (2131)  (2312)
                                  (2311)  (2321)
                                  (3112)  (3122)
                                  (3121)  (3212)
                                  (3211)  (3221)

Looking at run-lengths instead of multiplicities gives A329749.
The non-complete version is A329741.
Complete compositions are A107429.
Cf. A008965, A098504, A244164, A274174, A329740, A329744, A329766.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[Sort[#]]]&]],{n,0,10}]

a(21)-a(38) from Alois P. Heinz, Jul 06 2020

A333630 Least STC-number of a composition whose sequence of run-lengths has STC-number n.

Original entry on oeis.org

0, 1, 3, 5, 7, 14, 11, 13, 15, 30, 43, 29, 23, 46, 27, 45, 31, 62, 122, 61, 87, 117, 59, 118, 47, 94, 107, 93, 55, 110, 91, 109, 63, 126, 250, 125, 343, 245, 123, 246, 175, 350, 235, 349, 119, 238, 347, 237, 95, 190, 378, 189, 215, 373, 187, 374, 111, 222, 363

Offset: 0

Gus Wiseman, Mar 31 2020

nonn

All terms belong to A003754.

A composition of n is a finite sequence of positive integers summing to n. The composition with STC-number k (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
   0: ()
   1: (1)
   3: (1,1)
   5: (2,1)
   7: (1,1,1)
  14: (1,1,2)
  11: (2,1,1)
  13: (1,2,1)
  15: (1,1,1,1)
  30: (1,1,1,2)
  43: (2,2,1,1)
  29: (1,1,2,1)
  23: (2,1,1,1)
  46: (2,1,1,2)
  27: (1,2,1,1)
  45: (2,1,2,1)
  31: (1,1,1,1,1)
  62: (1,1,1,1,2)

Position of first appearance of n in A333627.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- Compositions without terms > 2 are A003754.
- Compositions without ones are ranked by A022340.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Runs-resistance is A333628.
- First appearances of run-resistances are A333629.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
seq=Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,1000}];
Table[Position[seq,i][[1,1]],{i,First[Split[Union[seq],#1+1==#2&]]}]-1

A003262 Let y=f(x) satisfy F(x,y)=0. a(n) is the number of terms in the expansion of (d/dx)^n y in terms of the partial derivatives of F.

Original entry on oeis.org

1, 3, 9, 24, 61, 145, 333, 732, 1565, 3247, 6583, 13047, 25379, 48477, 91159, 168883, 308736, 557335, 994638, 1755909, 3068960, 5313318, 9118049, 15516710, 26198568, 43904123, 73056724, 120750102, 198304922, 323685343

Offset: 1

N. J. A. Sloane

			(d/dx)^2 y = -F_xx/F_y + 2*F_x*F_xy/F_y^2 - F_x^2*F_yy/F_y^3, where F_x denotes partial derivative with respect to x, etc. This has three terms, thus a(2)=3.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 175.
L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..500
L. Comtet, Letter to N. J. A. Sloane, Mar 1974.
L. Comtet and M. Fiolet, Number of terms in an nth derivative, C. R. Acad. Sc. Paris, t. 278 (21 janvier 1974), Serie A- 249-251. (Annotated scanned copy)
T. Wilde, Implicit higher derivatives and a formula of Comtet and Fiolet, arXiv:0805.2674 [math.CO], 2008.

Cf. A098504.
Cf. A172004 (generalization to bivariate implicit functions).
Cf. A162326 (analogous sequence for implicit divided differences).
Cf. A172003 (bivariate variant).

p[, ] = 0; q[, ] = 0; e = 30; For[m = 1, m <= e - 1, m++, For[d = 1, d <= m, d++, If[m == d*Floor[m/d], For[i = 0, i <= m/d + 1, i++, If[d*i <= e, q[m, i*d] = q[m, i*d] + 1/d]]]]]; For[j = 0, j <= e, j++, p[0, j] = 1]; For[n = 1, n <= e - 1, n++, For[s = 0, s <= n, s++, For[j = 0, j <= e, j++, For[i = 0, i <= j, i++, p[n, j] = p[n, j] + (1/n)*s*q[s, j - i]*p[n - s, i]]]]]; A003262 = Table[p[n - 1, n], {n, 1, e}](* Jean-François Alcover, after Tom Wilde *)

' Tom Wilde, Jan 19 2008
Sub Calc_AofN_upto_E()
E = 30
ReDim p(0 To E - 1, 0 To E)
ReDim q(0 To E - 1, 0 To E)
For m = 1 To E - 1
  For d = 1 To m
    If m = d * Int(m / d) Then
      For i = 0 To m / d + 1
        If d * i <= E Then q(m, i * d) = q(m, i * d) + 1 / d
      Next
    End If
  Next
Next
For j = 0 To E
  p(0, j) = 1
Next
For n = 1 To E - 1
  For s = 0 To n
    For j = 0 To E
      For i = 0 To j
        p(n, j) = p(n, j) + 1 / n * s * q(s, j - i) * p(n - s, i)
      Next
    Next
  Next
Next
For n = 1 To E
   Debug.Print p(n - 1, n)
Next
End Sub

The generating function given by Comtet and Fiolet is incorrect.

a(n) = coefficient of t^n*u^(n-1) in Product_{i,j>=0,(i,j)<>(0,1)} (1 - t^i*u^(i+j-1))^(-1). - Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008

More terms from Tom Wilde (tom(AT)beech84.fsnet.co.uk), Jan 19 2008

A329749 Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 2, 3, 5, 11, 23, 40, 80, 180, 344, 661, 1321, 2657, 5268, 10481, 20903, 41572, 82734, 164998, 328304, 654510, 1305421, 2598811, 5182174, 10332978, 20594318, 41066611, 81897091, 163309679, 325707492, 649648912, 1295827380, 2584941276, 5156774487

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.

			The a(0) = 1 through a(6) = 11 compositions (empty column not shown):
  ()  (1)  (1,2)  (1,1,2)  (1,2,2)    (1,2,3)
           (2,1)  (1,2,1)  (2,1,2)    (1,3,2)
                  (2,1,1)  (2,2,1)    (2,1,3)
                           (1,1,2,1)  (2,3,1)
                           (1,2,1,1)  (3,1,2)
                                      (3,2,1)
                                      (1,2,1,2)
                                      (1,2,2,1)
                                      (2,1,1,2)
                                      (2,1,2,1)
                                      (1,1,2,1,1)

Looking at  multiplicities instead of run-lengths gives A329748.
The non-complete version is A329766.
Complete compositions are A107429.
Cf. A000740, A008965, A098504, A244164, A274174, A329738, A329739, A329740, A329741, A329744.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[#]]&]],{n,0,10}]

a(21)-a(35) from Alois P. Heinz, Jul 06 2020

A329869 Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 4, 5, 11, 19, 36, 77, 138, 252, 528, 1072, 2204, 4634, 9575, 19732, 40754

Offset: 0

Gus Wiseman, Nov 23 2019

A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(1) = 1 through a(9) = 19 compositions:
  1   2   3   4   5   6      7       8        9
      11      22      33     11113   44       11115
                      11112  31111   11114    12222
                      21111  111211  41111    22221
                             112111  111122   51111
                                     111311   111222
                                     113111   111411
                                     211112   114111
                                     221111   211113
                                     1111121  222111
                                     1211111  311112
                                              1111131
                                              1111221
                                              1112112
                                              1121112
                                              1221111
                                              1311111
                                              2111211
                                              2112111
For example, the runs-resistance of (1221111) is 3 because we have: (1221111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have: (1221111) -> (2111) -> (11) -> (1) -> (), so (1221111) is counted under a(9).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

The version for binary indices is A329866.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329864, A329865, A329867, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]+1==degdep[#]&]],{n,0,10}]

A325550 Number of necklace compositions of n with distinct multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 18, 41, 86, 118, 273, 465, 731, 1432, 2791, 4063, 8429, 14761, 29465, 58654, 123799, 227419, 453229, 861909, 1697645, 3192807, 6315007, 11718879, 22795272, 42965245, 83615516, 156215020, 306561088, 587300503, 1140650287, 2203107028

Offset: 1

Gus Wiseman, May 10 2019

nonn

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

			The a(1) = 1 through a(8) = 16 necklace compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (113)    (33)      (115)      (44)
                    (112)   (122)    (114)     (133)      (116)
                    (1111)  (1112)   (222)     (223)      (224)
                            (11111)  (1113)    (1114)     (233)
                                     (11112)   (1222)     (1115)
                                     (111111)  (11113)    (2222)
                                               (11122)    (11114)
                                               (11212)    (11222)
                                               (111112)   (12122)
                                               (1111111)  (111113)
                                                          (111122)
                                                          (111212)
                                                          (112112)
                                                          (1111112)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000079, A000740, A008965, A059966, A098504, A098859, A242882, A242887, A325549, A325554.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Length/@Split[Sort[#]]&]],{n,15}]

b(n)={((r,k,b,w)->if(!k||!r, if(r,0,(w-1)!), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b,1<Andrew Howroyd, Aug 31 2019

a(n) = Sum_{d|n} phi(d)*(Sum_{k=1..n/d} A242887(n/d, k)/k)/d. - Andrew Howroyd, Aug 31 2019

Terms a(26) and beyond from Andrew Howroyd, Aug 31 2019

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A329741 Number of compositions of n whose multiplicities cover an initial interval of positive integers.

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A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.

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A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

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A333629 Least k such that the runs-resistance of the k-th composition in standard order is n.

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A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

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A333630 Least STC-number of a composition whose sequence of run-lengths has STC-number n.

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A003262 Let y=f(x) satisfy F(x,y)=0. a(n) is the number of terms in the expansion of (d/dx)^n y in terms of the partial derivatives of F.

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A329749 Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.

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