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A022811
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Number of terms in n-th derivative of a function composed with itself 3 times.
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15
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 non-increasing, 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 3, 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. [ From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 12 2008]
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REFERENCES
| W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
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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).
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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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 12 2008]
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CROSSREFS
| Cf. A008778, A022812-A022818, A024207-A024210. First column of A039805.
A row or column of A081718.
Sequence in context: A048134 A058397 A174369 * A002799 A162426 A058554
Adjacent sequences: A022808 A022809 A022810 * A022812 A022813 A022814
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KEYWORD
| nonn
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AUTHOR
| Winston C. Yang (yang(AT)math.wisc.edu)
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