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A022818
Square array read by antidiagonals: A(n,k) = number of terms in the n-th derivative of a function composed with itself k times (n, k >= 1).
18
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 5, 1, 1, 5, 10, 13, 7, 1, 1, 6, 15, 26, 23, 11, 1, 1, 7, 21, 45, 55, 44, 15, 1, 1, 8, 28, 71, 110, 121, 74, 22, 1, 1, 9, 36, 105, 196, 271, 237, 129, 30, 1, 1, 10, 45, 148, 322, 532, 599, 468, 210, 42, 1, 1, 11, 55, 201, 498, 952, 1301, 1309, 867, 345, 56, 1
OFFSET
1,5
REFERENCES
Winston C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.
LINKS
Alois P. Heinz, Antidiagonals n = 1..141
Warren P. Johnson, The curious history of Faà di Bruno's formula, American Mathematical Monthly, 109 (2002), 217-234.
Warren P. Johnson, The curious history of Faà di Bruno's formula, American Mathematical Monthly, 109 (2002), 217-234.
Winston C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245. [Take the transpose of Table 2 on p. 241 and omit row 0 and column 0; A(n,k) = M(k,n). - Petros Hadjicostas, May 30 2020]
FORMULA
From Petros Hadjicostas, May 30 2020: (Start)
A(n,k) = Sum_{s=1..n} A008284(n,s)*A(s,k-1) for n >= 1 and k >= 2 with A(n,1) = 1 for n >= 1.
A(n,k) = Sum_{s=1..n} binomial(k,s-1)*A081719(n-1,s-1) for n, k >= 1. (End)
EXAMPLE
Square array A(n,k) (with rows n >= 1 and columns k >= 1) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
1, 3, 6, 10, 15, 21, 28, 36, ...
1, 5, 13, 26, 45, 71, 105, 148, ...
1, 7, 23, 55, 110, 196, 322, 498, ...
1, 11, 44, 121, 271, 532, 952, 1590, ...
1, 15, 74, 237, 599, 1301, 2541, 4586, ...
1, 22, 129, 468, 1309, 3101, 6539, 12644, ...
...
MAPLE
A:= proc(n, k) option remember;
`if`(k=1, 1, add(b(n, n, i)*A(i, k-1), i=0..n))
end:
b:= proc(n, i, k) option remember; `if`(n<k, 0, `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1, k-j), j=0..min(n/i, k)))))
end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12); # Alois P. Heinz, Aug 18 2012
# second Maple program:
b:= proc(n, i, l, k) option remember; `if`(k=0,
`if`(n<2, 1, 0), `if`(n=0 or i=1, b(l+n$2, 0, k-1),
b(n, i-1, l, k) +b(n-i, min(n-i, i), l+1, k)))
end:
A:= (n, k)-> b(n$2, 0, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12); # Alois P. Heinz, Jul 19 2018
MATHEMATICA
a[n_, k_] := a[n, k] = If[k == 1, 1, Sum[b[n, n, i]*a[i, k-1], {i, 0, n}]]; 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]}]]]]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)
PROG
(PARI) P(n, k) = #partitions(n-k, k); /* A008284 */
tabl(nn) = {M = matrix(nn, nn, n, k, 0); for(n=1, nn, M[n, 1] = 1; ); for(n=1, nn, for(k=2, nn, M[n, k] = sum(s=1, n, P(n, s)*M[s, k-1]))); for (n=1, nn, for (k=1, nn, print1(M[n, k], ", "); ); print(); ); } \\ Petros Hadjicostas, May 30 2020
CROSSREFS
Main diagonal gives: A192435.
Sequence in context: A073714 A171848 A144151 * A050447 A248601 A167172
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited by Alois P. Heinz, Aug 18 2012
STATUS
approved