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A022818 Square array read by antidiagonals: A(n,k) = number of terms in n-th derivative of a function composed with itself k times. 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

REFERENCES

W. C. Yang (yang(AT)math.wisc.edu), Derivatives of self-compositions of functions, preprint, 1997.

LINKS

Alois P. Heinz, Antidiagonals n = 1..141

W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

EXAMPLE

Square array 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-Fran├žois Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

CROSSREFS

Rows n=1-10 give: A000012, A000027, A000217, A008778, A022815, A022816, A022817, A215626, A215627, A215628.

Columns k=1-10 give: A000012, A000041, A022811, A022812, A022813, A022814, A024207, A024208, A024209, A024210.

Main diagonal gives: A192435.

Sequence in context: A073714 A171848 A144151 * A050447 A248601 A167172

Adjacent sequences:  A022815 A022816 A022817 * A022819 A022820 A022821

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by Alois P. Heinz, Aug 18 2012

STATUS

approved

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Last modified February 19 22:38 EST 2019. Contains 320328 sequences. (Running on oeis4.)