A022811 -id:A022811 - cp's OEIS frontend

A133494 Diagonal of the array of iterated differences of A047848.

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A008778 a(n) = (n+1)(n^2 +8n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 5, 13, 26, 45, 71, 105, 148, 201, 265, 341, 430, 533, 651, 785, 936, 1105, 1293, 1501, 1730, 1981, 2255, 2553, 2876, 3225, 3601, 4005, 4438, 4901, 5395, 5921, 6480, 7073, 7701, 8365, 9066, 9805, 10583, 11401, 12260, 13161, 14105, 15093, 16126, 17205, 18331

Offset: 0

N. J. A. Sloane

Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i-1,j)-m(i-1,j-1); then a(n)=m(n+3,3) - Benoit Cloitre, May 08 2002
a(n) = number of (n+6)-bit binary sequences with exactly 6 1's none of which is isolated. - David Callan, Jul 15 2004
If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 03 2007
Sum of first n triangular numbers plus previous triangular number. - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
a(n) = Sum of first (n+1) triangular numbers plus n-th triangular number (see penultimate formula by Henry Bottomley). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
For n > 0, a(n-1) is the number of compositions of n+6 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
The binomial transform of [1,4,4,1,0,0,0,...], the 4th row in A116672. - R. J. Mathar, Jul 18 2017

			G.f. = 1 + 5*x + 13*x^2 + 26*x^3 + 45*x^4 + 71*x^5 + 105*x^6 + 148*x^7 + 201*x^8 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190 eq. (11.4.7).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
Francisco Javier de Vega, Some Variants of Integer Multiplication, Axioms (2023) Vol. 12, 905. See p. 15.
Milan Janjic, Two Enumerative Functions
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A022811-A022817, A024207-A024210.
Column 1 of triangle A094415.
Row n=4 of A022818.
Cf. A002411, A008779, A005712 (partial sums), A034856 (first diffs).

List([0..50], n-> (n+1)*(n^2 +8*n +6)/6); # G. C. Greubel, Sep 11 2019

[(n+1)*(n^2+8*n+6)/6: n in [0..50]]; // Vincenzo Librandi, May 21 2011

seq(1+4*k+4*binomial(k, 2)+binomial(k, 3), k=0..45);

Table[(n+1)*(n^2+8*n+6)/6, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009, modified by G. C. Greubel, Sep 11 2019 *)
LinearRecurrence[{4,-6,4,-1}, {1,5,13,26}, 51] (* G. C. Greubel, Sep 11 2019 *)

Vec((1+x-x^2)/(1-x)^4 + O(x^50)) \\ Altug Alkan, Jan 07 2016

[(n+1)*(n^2 +8*n +6)/6 for n in (0..50)] # G. C. Greubel, Sep 11 2019

a(n) = dot_product(n, n-1, ...2, 1)*(2, 3, ..., n, 1) for n = 2, 3, 4, ... [i.e., a(2) = (2, 1)*(2, 1), a(3) = (3, 2, 1)*(2, 3, 1)]. - Clark Kimberling
a(n) = a(n-1) + A034856(n+1) = A000297(n-1) + 1 = A000217(n) + A000292(n+1) = A000290(n-1) + A000292(n). - Henry Bottomley, Oct 25 2001
a(n) = Sum_{0<=k, l<=n; k+l|n} k*l. - Ralf Stephan, May 06 2005
G.f.: (1+x-x^2)/(1-x)^4. - Colin Barker, Jan 06 2012
a(n) = A000330(n+1) - A000292(n-1). - Bruce J. Nicholson, Jul 05 2018
E.g.f.: (6 +24*x +12*x^2 +x^3)*exp(x)/6. - G. C. Greubel, Sep 11 2019

A022818 Square array read by antidiagonals: A(n,k) = number of terms in the n-th derivative of a function composed with itself k times (n, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 13, 7, 1, 1, 6, 15, 26, 23, 11, 1, 1, 7, 21, 45, 55, 44, 15, 1, 1, 8, 28, 71, 110, 121, 74, 22, 1, 1, 9, 36, 105, 196, 271, 237, 129, 30, 1, 1, 10, 45, 148, 322, 532, 599, 468, 210, 42, 1, 1, 11, 55, 201, 498, 952, 1301, 1309, 867, 345, 56, 1

Offset: 1

N. J. A. Sloane

			Square array A(n,k) (with rows n >= 1 and columns k >= 1) begins:
  1,  1,   1,   1,    1,    1,    1,     1, ...
  1,  2,   3,   4,    5,    6,    7,     8, ...
  1,  3,   6,  10,   15,   21,   28,    36, ...
  1,  5,  13,  26,   45,   71,  105,   148, ...
  1,  7,  23,  55,  110,  196,  322,   498, ...
  1, 11,  44, 121,  271,  532,  952,  1590, ...
  1, 15,  74, 237,  599, 1301, 2541,  4586, ...
  1, 22, 129, 468, 1309, 3101, 6539, 12644, ...
  ...

Winston C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Alois P. Heinz, Antidiagonals n = 1..141
Warren P. Johnson, The curious history of Faà di Bruno's formula, American Mathematical Monthly, 109 (2002), 217-234.
Warren P. Johnson, The curious history of Faà di Bruno's formula, American Mathematical Monthly, 109 (2002), 217-234.
Winston C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245. [Take the transpose of Table 2 on p. 241 and omit row 0 and column 0; A(n,k) = M(k,n). - _Petros Hadjicostas_, May 30 2020]

Rows n=1-10 give: A000012, A000027, A000217, A008778, A022815, A022816, A022817, A215626, A215627, A215628.
Columns k=1-10 give: A000012, A000041, A022811, A022812, A022813, A022814, A024207, A024208, A024209, A024210.
Main diagonal gives: A192435.
Cf. A008284, A081719.

A:= proc(n, k) option remember;
      `if`(k=1, 1, add(b(n, n, i)*A(i, k-1), i=0..n))
    end:
b:= proc(n, i, k) option remember; `if`(nAlois P. Heinz, Aug 18 2012
# second Maple program:
b:= proc(n, i, l, k) option remember; `if`(k=0,
      `if`(n<2, 1, 0), `if`(n=0 or i=1, b(l+n$2, 0, k-1),
         b(n, i-1, l, k) +b(n-i, min(n-i, i), l+1, k)))
    end:
A:= (n, k)->  b(n$2, 0, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Jul 19 2018

a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]]; b[n_, i_, k_] := b[n, i, k] = If[n < k, 0, If[n == 0, 1, If[i < 1, 0, Sum[b[n-i*j, i-1, k-j], {j, 0, Min[n/i, k]}]]]]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

P(n, k) = #partitions(n-k, k); /* A008284 */
tabl(nn) = {M = matrix(nn, nn, n, k, 0); for(n=1, nn, M[n, 1] = 1; ); for(n=1, nn, for(k=2, nn, M[n, k] = sum(s=1, n, P(n, s)*M[s, k-1]))); for (n=1, nn, for (k=1, nn, print1(M[n, k], ", "); ); print(); ); } \\ Petros Hadjicostas, May 30 2020

From Petros Hadjicostas, May 30 2020: (Start)
A(n,k) = Sum_{s=1..n} A008284(n,s)*A(s,k-1) for n >= 1 and k >= 2 with A(n,1) = 1 for n >= 1.
A(n,k) = Sum_{s=1..n} binomial(k,s-1)*A081719(n-1,s-1) for n, k >= 1. (End)

Edited by Alois P. Heinz, Aug 18 2012

A024207 Number of terms in n-th derivative of a function composed with itself 7 times.

Original entry on oeis.org

1, 1, 7, 28, 105, 322, 952, 2541, 6539, 15833, 37148, 83594, 183289, 389520, 809820, 1643375, 3272797, 6390745, 12279337, 23208483, 43252360, 79483096, 144265338, 258673983, 458747540, 804877837, 1398356706, 2406328974, 4104352128, 6940717598, 11643270856

Offset: 0

Winston C. Yang (yang(AT)math.wisc.edu)

nonn

W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Cf. A008778, A022811-A022817, A024208-A024210. First column of A050301.

Column k=7 of A022818.

b[n_, i_, k_] := b[n, i, k] = If[n < k, 0, If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k - j], {j, 0, Min[n/i, k]}]]]];
a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
a[n_] := a[n, 7];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)

If a(n,m) = number of terms in m-derivative of a function composed with itself n times, p(n,k) = number of partitions of n into k parts, then a(n,m) = sum_{i=0..m} p(m,i)*a(n-1,i).

More terms from Alois P. Heinz, Aug 18 2012

A024210 Number of terms in n-th derivative of a function composed with itself 10 times.

Original entry on oeis.org

1, 1, 10, 55, 265, 1045, 3817, 12583, 39148, 114235, 318857, 850576, 2190850, 5451721, 13184711, 31023842, 71286349, 160139911, 352574213, 761567304, 1616713932, 3376143283, 6944345483, 14080091227, 28169087367, 55644767253, 108617341172, 209626751905

Offset: 0

Winston C. Yang (yang(AT)math.wisc.edu)

nonn

W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Cf. A008778, A022811-A022817, A024207-A024209. First column of A050304.

Column k=10 of A022818.

b[n_, i_, k_] := b[n, i, k] = If[n < k, 0, If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k - j], {j, 0, Min[n/i, k]}]]]];
a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]];
a[n_] := a[n, 10]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)

If a(n,m) = number of terms in m-derivative of a function composed with itself n times, p(n,k) = number of partitions of n into k parts, then a(n,m) = sum_{i=0..m} p(m,i)*a(n-1,i).

A038497 Matrix square of partition triangle A008284.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 7, 8, 5, 2, 1, 11, 15, 10, 5, 2, 1, 15, 23, 18, 10, 5, 2, 1, 22, 38, 31, 20, 10, 5, 2, 1, 30, 56, 52, 34, 20, 10, 5, 2, 1, 42, 86, 83, 60, 36, 20, 10, 5, 2, 1, 56, 123, 129, 97, 63, 36, 20, 10, 5, 2, 1, 77, 181, 198, 158, 105, 65, 36, 20, 10, 5, 2, 1

Offset: 1

Christian G. Bower, Feb 15 1999

Row sums form A022811 (number of terms in n-th derivative of a function composed with itself 3 times). - Paul D. Hanna, Jul 13 2004

			1; 2,1; 3,2,1; 5,5,2,1; ...

Cf. A038498, A039800-A039809. a(n, 1) = A000041(n) (first column) (partition numbers).

Cf. A022811.

A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 4, 12, 30, 66, 164, 419, 1049, 2625, 6372, 15451, 37335, 89855, 216523, 518714, 1235897, 2930050, 6911149, 16217817, 37914515, 88304358, 204971388, 474172899, 1093547574, 2513959446, 5761735383, 13165908506, 29998936859, 68164839887, 154478212575

Offset: 0

Gus Wiseman, Jul 01 2022

nonn

If a composition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of compositions.

			The initial terms count the following containments:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)
                (11)(1)  (21)(21)  (31)(31)
                         (12)(12)  (13)(13)
                         (111)(1)  (22)(2)
                                   (211)(11)
                                   (211)(21)
                                   (121)(11)
                                   (121)(12)
                                   (121)(21)
                                   (112)(11)
                                   (112)(12)
                                   (1111)(1)

Christian Sievers, Table of n, a(n) for n = 0..59

The homog. case is A032020, w/o containment A355388 (partitions A355385).
For partitions we have A355383, homog. A000009, w/ multiplicity A339006.
A000244 counts splittings of compositions, for partitions A323583.
Cf. A001970, A022811, A063834, A070933, A072706, A133494, A336139.

Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,5}]

a(21) and beyond from Christian Sievers, May 08 2025

A039805 Matrix cube of partition triangle A008284.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 13, 9, 3, 1, 23, 19, 9, 3, 1, 44, 42, 22, 9, 3, 1, 74, 80, 48, 22, 9, 3, 1, 129, 154, 99, 51, 22, 9, 3, 1, 210, 273, 193, 105, 51, 22, 9, 3, 1, 345, 484, 362, 212, 108, 51, 22, 9, 3, 1, 542, 815, 651, 401, 218, 108, 51, 22, 9, 3, 1, 858, 1369, 1147

Offset: 1

Christian G. Bower, Feb 15 1999

Row sums form A022812 (number of terms in n-th derivative of a function composed with itself 4 times). - Paul D. Hanna, Jul 13 2004

			1; 3,1; 6,3,1; 13,9,3,1; ...

Cf. A038497, A038498, A039806, A039807. a(n, 1) = A022811(n) (first column).

Cf. A022812.

A022817 Number of terms in 7th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 15, 74, 237, 599, 1301, 2541, 4586, 7785, 12583, 19536, 29327, 42783, 60893, 84827, 115956, 155873, 206415, 269686, 348081, 444311, 561429, 702857, 872414, 1074345, 1313351, 1594620, 1923859, 2307327, 2751869, 3264951, 3854696, 4529921, 5300175, 6175778

Offset: 1

N. J. A. Sloane

nonn

W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Cf. A008778, A022811-A022816, A024207-A024210.

Row n=4 of A022818.

a:= n-> n*(36+(-356+(645+(355+(39+n)*n)*n)*n)*n)/720:
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 18 2012

Table[(n/720*(n^5+39*n^4+355*n^3+645*n^2-356*n+36)),{n,1,100}] (* Vincenzo Librandi, Aug 18 2012 *)

a(n) = n/720 * (n^5 + 39*n^4 + 355*n^3 + 645*n^2 - 356*n + 36).

G.f.: (x^5-4*x^4+x^3+10*x^2-8*x-1)*x/(x-1)^7. - Alois P. Heinz, Aug 18 2012

More terms from Christian G. Bower, Aug 15 1999.

A131408 Repeated integer partitions or nested integer partitions.

Original entry on oeis.org

1, 1, 2, 5, 14, 35, 95, 248, 668, 1781, 4799, 12890, 34766, 93647, 252635, 681272, 1838135, 4958738, 13379885, 36100214, 97409045, 262833314, 709207394, 1913652308, 5163654671, 13933178390, 37596275726, 101446960109, 273737216768, 738632652929, 1993073801930

Offset: 0

Thomas Wieder, Jul 09 2007

nonn

See A131407 for the labeled case (with much more explanation).

Also the number of sequences of distinct integer partitions (y_1, ..., y_k), containing no partitions of the form (111..1) other than (1), such that sum(y_1) = n and length(y_i) = sum(y_{i+1}) for all i = 1, ..., k-1. - Gus Wiseman, Jul 20 2018

			Let denote * an unlabeled element. Then a(n=3)=5 because we have [ *,*,* ], [ *, * ][ * ], [[ *,* ]][[ * ]], [[ *,* ][ * ]], [ * ][ * ][ * ].
From _Gus Wiseman_, Jul 20 2018: (Start)
The a(4) = 14 sequences of integer partitions:
  (4), (31), (22), (211),
  (4)(1), (31)(2), (22)(2), (211)(3), (211)(21),
  (31)(2)(1), (22)(2)(1), (211)(3)(1), (211)(21)(2),
  (211)(21)(2)(1).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2321
Thomas Wieder, Visual Basic Program

Cf. A022811, A022818, A131407, A246828.

A000041 := proc(n) combinat[numbpart](n) ; end: A008284 := proc(n,k) if k = 1 or k = n then 1; elif k > n then 0 ; else procname(n-1,k-1)+procname(n-k,k) ; fi ; end: A131408 := proc(n) option remember; local i ; if n <= 2 then n; else A000041(n)+add(A008284(n,i)*procname(i),i=2..n-1) ; fi ; end: for n from 1 to 40 do printf("%d,",A131408(n)) ; od: # R. J. Mathar, Aug 07 2008
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) + b(n-i, min(n-i, i)))
    end:
a:= proc(n) option remember; b(n$2)+
      add(b(n-i, min(n-i, i))*a(i), i=2..n-1)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 03 2020

t[, 1] = 1; t[n, k_] /; 1 <= k <= n := t[n, k] = Sum[t[n-i, k-1], {i, 1, n-1}] - Sum[t[n-i, k], {i, 1, k-1}]; t[, ] = 0; a[1]=1; a[2]=2; a[n_] := a[n] = PartitionsP[n] + Sum[t[n, i]*a[i], {i, 2, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 02 2017 *)
roo[n_]:=If[n==1,{{{1}}},Join@@Cases[Most[IntegerPartitions[n]],y_:>Prepend[(Prepend[#,y]&/@roo[Length[y]]),{y}]]];
Table[Length[roo[n]],{n,10}] (* Gus Wiseman, Jul 20 2018 *)

a(n) = A000041(n) + Sum_{i=2..n-1} A008284(n,i)*a(i).

a(n) ~ c * d^n, where d = A246828 = 2.69832910647421123126399866618837633..., c = 0.232635324064951140265176690908381464098550827908380222089145... . - Vaclav Kotesovec, Sep 04 2014

Edited and extended by R. J. Mathar, Aug 07 2008

a(0)=1 prepended and edited by Alois P. Heinz, Sep 03 2020

cp's OEIS Frontend

A133494 Diagonal of the array of iterated differences of A047848.

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A008778 a(n) = (n+1)*(n^2 +8*n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.

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A022818 Square array read by antidiagonals: A(n,k) = number of terms in the n-th derivative of a function composed with itself k times (n, k >= 1).

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A024207 Number of terms in n-th derivative of a function composed with itself 7 times.

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A024210 Number of terms in n-th derivative of a function composed with itself 10 times.

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A038497 Matrix square of partition triangle A008284.

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A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.

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A008778 a(n) = (n+1)(n^2 +8n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.