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 A022812 Number of terms in n-th derivative of a function composed with itself 4 times. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997. LINKS 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. 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). MATHEMATICA 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 *) CROSSREFS Cf. A008778, A022811-A022818, A024207-A024210. First column of A039806. Sequence in context: A269064 A307415 A322060 * A192306 A276432 A000293 Adjacent sequences:  A022809 A022810 A022811 * A022813 A022814 A022815 KEYWORD nonn AUTHOR Winston C. Yang (yang(AT)math.wisc.edu) STATUS approved

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Last modified June 26 10:12 EDT 2019. Contains 324375 sequences. (Running on oeis4.)