A039806 -id:A039806 - cp's OEIS frontend

A039805 Matrix cube of partition triangle A008284.

1, 3, 1, 6, 3, 1, 13, 9, 3, 1, 23, 19, 9, 3, 1, 44, 42, 22, 9, 3, 1, 74, 80, 48, 22, 9, 3, 1, 129, 154, 99, 51, 22, 9, 3, 1, 210, 273, 193, 105, 51, 22, 9, 3, 1, 345, 484, 362, 212, 108, 51, 22, 9, 3, 1, 542, 815, 651, 401, 218, 108, 51, 22, 9, 3, 1, 858, 1369, 1147

Offset: 1

Christian G. Bower, Feb 15 1999

Row sums form A022812 (number of terms in n-th derivative of a function composed with itself 4 times). - Paul D. Hanna, Jul 13 2004

			1; 3,1; 6,3,1; 13,9,3,1; ...

Cf. A038497, A038498, A039806, A039807. a(n, 1) = A022811(n) (first column).

Cf. A022812.

A039807 Matrix 5th power of partition triangle A008284.

Original entry on oeis.org

1, 5, 1, 15, 5, 1, 45, 20, 5, 1, 110, 60, 20, 5, 1, 271, 170, 65, 20, 5, 1, 599, 426, 185, 65, 20, 5, 1, 1309, 1025, 486, 190, 65, 20, 5, 1, 2690, 2299, 1185, 501, 190, 65, 20, 5, 1, 5436, 4999, 2750, 1245, 506, 190, 65, 20, 5, 1, 10545, 10380, 6069, 2910, 1260

Offset: 0

Christian G. Bower, Feb 15 1999

			1; 5,1; 15,5,1; 45,20,5,1; ...

Cf. A038497, A038498, A039805, A039806. a(n, 1) = A022813(n) (first column).

A022812 Number of terms in n-th derivative of a function composed with itself 4 times.

Original entry on oeis.org

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

Offset: 0

Winston C. Yang (yang(AT)math.wisc.edu)

nonn

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

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.

Cf. A008778, A022811-A022818, A024207-A024210. First column of A039806.

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 *)

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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A039805 Matrix cube of partition triangle A008284.

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A039807 Matrix 5th power of partition triangle A008284.

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A022812 Number of terms in n-th derivative of a function composed with itself 4 times.

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