login
A024210
Number of terms in n-th derivative of a function composed with itself 10 times.
14
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
0,3
REFERENCES
W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
LINKS
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, 10]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 28 2017, after Alois P. Heinz *)
CROSSREFS
Cf. A008778, A022811-A022817, A024207-A024209. First column of A050304.
Column k=10 of A022818.
Sequence in context: A202481 A348663 A169720 * A253002 A357668 A316109
KEYWORD
nonn
AUTHOR
Winston C. Yang (yang(AT)math.wisc.edu)
STATUS
approved