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%I #43 Jan 28 2020 16:09:32
%S 1,1,3,6,13,23,44,74,129,210,345,542,858,1310,2004,2996,4467,6540,
%T 9552,13744,19711,27943,39452,55172,76865,106200,146173,199806,272075,
%U 368247,496642,666201,890602,1184957,1571417,2075058,2731677,3582119,4683595,6102256
%N Number of terms in n-th derivative of a function composed with itself 3 times.
%C 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
%C 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
%D W. C. Yang, Derivatives of self-compositions of functions, preprint, 1997.
%H Alois P. Heinz, <a href="/A022811/b022811.txt">Table of n, a(n) for n = 0..3000</a> (terms n = 501..959 from Vaclav Kotesovec)
%H W. C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Mathematics, 222(1-3), July 2000, 235-245.
%F 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).
%F 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
%e From _Gus Wiseman_, Jul 19 2018: (Start)
%e Using the chain rule, we compute the second derivative of f(f(f(x))) to be the following sum of a(2) = 3 terms.
%e d^2/dx^2 f(f(f(x))) =
%e f'(f(x)) f'(f(f(x))) f''(x) +
%e f'(x)^2 f'(f(f(x))) f''(f(x)) +
%e f'(x)^2 f'(f(x))^2 f''(f(f(x))).
%e (End)
%p 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
%t a[n_] := Total[PartitionsP[Length[#]]& /@ IntegerPartitions[n]];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 80}] (* _Jean-François Alcover_, Apr 28 2017 *)
%t Table[Length[1+D[f[f[f[x]]],{x,n}]]-1,{n,10}] (* _Gus Wiseman_, Jul 19 2018 *)
%Y Column k=3 of A022818.
%Y First column of A039805.
%Y A row or column of A081718.
%Y Cf. A008778, A022812, A022813, A022814, A022815, A022816, A022817, A024207, A024208, A024209, A024210, A131408.
%K nonn
%O 0,3
%A Winston C. Yang (yang(AT)math.wisc.edu)
%E Typo corrected by Neven Juric, Mar 25 2013
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