A024207 -id:A024207 - cp's OEIS frontend

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.

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

A022811 Number of terms in n-th derivative of a function composed with itself 3 times.

Original entry on oeis.org

1, 1, 3, 6, 13, 23, 44, 74, 129, 210, 345, 542, 858, 1310, 2004, 2996, 4467, 6540, 9552, 13744, 19711, 27943, 39452, 55172, 76865, 106200, 146173, 199806, 272075, 368247, 496642, 666201, 890602, 1184957, 1571417, 2075058, 2731677, 3582119, 4683595, 6102256

Offset: 0

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

nonn

This also counts a restricted set of plane partitions of n. Each element of the set which contains the A000041(n) partitions of n can be converted into plane partitions by insertion of line feeds at some or all places of the "pluses." Since the length of rows in plane partitions must be nonincreasing, there are only A000041(L(P)) ways to comply with this rule, where L(P) is the number of terms in that particular partition. Example for n=4: consider all five partitions 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1 of four. The associated a(4)=13 plane partitions are 4, 31, 3|1, 22, 2|2, 211, 21|1, 2|1|1, 1111, 111|1, 11|11, 11|1|1 and 1|1|1|1, where the bar represents start of the next row, where a(4) = A000041(L(4)) + A000041(L(3+1)) + A000041(L(2+2)) + A000041(L(2+1+1))+ A000041(L(1+1+1+1)) = A000041(1) + A000041(2) + A000041(2) + A000041(3) + A000041(4). By construction from sorted partitions, all the plane partitions are strictly decreasing along each row and each column. - R. J. Mathar, Aug 12 2008

Also the number of pairs of integer partitions, the first with sum n and the second with sum equal to the length of the first. - Gus Wiseman, Jul 19 2018

			From _Gus Wiseman_, Jul 19 2018: (Start)
Using the chain rule, we compute the second derivative of f(f(f(x))) to be the following sum of a(2) = 3 terms.
  d^2/dx^2 f(f(f(x))) =
  f'(f(x)) f'(f(f(x))) f''(x) +
  f'(x)^2 f'(f(f(x))) f''(f(x)) +
  f'(x)^2 f'(f(x))^2 f''(f(f(x))).
(End)

W. C. Yang, Derivatives of self-compositions of functions, preprint, 1997.

Alois P. Heinz, Table of n, a(n) for n = 0..3000 (terms n = 501..959 from Vaclav Kotesovec)
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Column k=3 of A022818.
First column of A039805.
A row or column of A081718.
Cf. A008778, A022812, A022813, A022814, A022815, A022816, A022817, A024207, A024208, A024209, A024210, A131408.

A022811 := proc(n) local a,P,p,lp ; a := 0 ; P := combinat[partition](n) ; for p in P do lp := nops(p) ; a := a+combinat[numbpart](lp) ; od: RETURN(a) ; end: for n from 1 do print(n,A022811(n)) ; od: # R. J. Mathar, Aug 12 2008

a[n_] := Total[PartitionsP[Length[#]]& /@ IntegerPartitions[n]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 80}] (* Jean-François Alcover, Apr 28 2017 *)
Table[Length[1+D[f[f[f[x]]],{x,n}]]-1,{n,10}] (* Gus Wiseman, Jul 19 2018 *)

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).

G.f.: Sum_{k>=0} p(k) * x^k / Product_{j=1..k} (1 - x^j), where p(k) = number of partitions of k. - Ilya Gutkovskiy, Jan 28 2020

Typo corrected by Neven Juric, Mar 25 2013

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

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).

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.

A022812 Number of terms in n-th derivative of a function composed with itself 4 times.

Original entry on oeis.org

1, 1, 4, 10, 26, 55, 121, 237, 468, 867, 1597, 2821, 4952, 8421, 14206, 23439, 38324, 61570, 98112, 154111, 240197, 370015, 565802, 856664, 1288366, 1921016, 2846572, 4186730, 6122369, 8893904, 12851713, 18460961, 26388354, 37519159, 53101687, 74792210

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-A022818, A024207-A024210. First column of A039806.

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, 4]; Table[a[n], {n, 0, 40}] (* 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).

A022813 Number of terms in n-th derivative of a function composed with itself 5 times.

Original entry on oeis.org

1, 1, 5, 15, 45, 110, 271, 599, 1309, 2690, 5436, 10545, 20148, 37341, 68223, 121878, 214846, 371993, 636570, 1073325, 1790721, 2950922, 4816603, 7778937, 12455988, 19761148, 31108121, 48572686, 75307513, 115909727, 177255526, 269294119, 406708721, 610593948

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-A022818, A024207-A024210. First column of A039807.

b[n_, i_, k_] := b[n, i, k] = If[nJean-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).

A022814 Number of terms in n-th derivative of a function composed with itself 6 times.

Original entry on oeis.org

1, 1, 6, 21, 71, 196, 532, 1301, 3101, 6956, 15217, 31951, 65670, 130914, 256150, 489690, 920905, 1699693, 3092751, 5540571, 9802091, 17114237, 29550346, 50444952, 85264328, 142682505, 236649524, 389033014, 634408230, 1026350152, 1648328017, 2628254619

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-A022818, A024207-A024210. First column of A050300.

b[n_, i_, k_] := b[n, i, k] = If[nJean-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).

A022815 Number of terms in 5th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 7, 23, 55, 110, 196, 322, 498, 735, 1045, 1441, 1937, 2548, 3290, 4180, 5236, 6477, 7923, 9595, 11515, 13706, 16192, 18998, 22150, 25675, 29601, 33957, 38773, 44080, 49910, 56296, 63272, 70873, 79135, 88095, 97791, 108262, 119548, 131690

Offset: 1

N. J. A. Sloane

			a(7) = 7*28 + (7*0+6*1+5*3+4*6+3*10+2*15+1*21) = 322. [_Bruno Berselli_, Jun 22 2013]

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

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Cf. A008778, A022811-A022818, A024207-A024210.

[n*(n+1)*(n^2+13*n-2)/24: n in [1..40]]; // Vincenzo Librandi, Oct 10 2011

a(n) = n*(n+1)*(n^2+13*n-2)/24. - John W. Layman, Apr 27 2000
G.f.: x*(1-2*x^2+2*x)/(1-x)^5. [Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]
a(n) = n*A000217(n) + sum((n-i)*A000217(i), i=0..n-1). [Bruno Berselli, Jun 23 2013]

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

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

A024208 Number of terms in n-th derivative of a function composed with itself 8 times.

Original entry on oeis.org

1, 1, 8, 36, 148, 498, 1590, 4586, 12644, 32775, 81901, 196085, 455772, 1025779, 2252674, 4823546, 10116553, 20783490, 41949270, 83211931, 162552093, 312850854, 594086542, 1113610526, 2062796698, 3777567977, 6844786250, 12276620372, 21809737429, 38391720375

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-A024210. First column of A050302.

Column k=8 of A022818.

b[n_, i_, k_] := b[n, i, k] = If[nJean-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).

cp's OEIS Frontend

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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A022811 Number of terms in n-th derivative of a function composed with itself 3 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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A024210 Number of terms in n-th derivative of a function composed with itself 10 times.

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

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A022812 Number of terms in n-th derivative of a function composed with itself 4 times.

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A022813 Number of terms in n-th derivative of a function composed with itself 5 times.

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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.