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

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

Author

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

Keywords

• nonn

Comments

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

Examples

			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)
		

References

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

Links

• 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.

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

Typo corrected by Neven Juric, Mar 25 2013