A024207 -id:A024207 - cp's OEIS frontend

A024209 Number of terms in n-th derivative of a function composed with itself 9 times.

1, 1, 9, 45, 201, 735, 2517, 7785, 22857, 63024, 166819, 422537, 1035971, 2456694, 5672347, 12756334, 28053280, 60371967, 127479247, 264311585, 539102751, 1082474167, 2142579168, 4183251750, 8064722973, 15360809911, 28928858208, 53896616704, 99398216733

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-A022817, A024207-A024210. First column of A050303.

Column k=9 of A022818.

b[n_, i_, k_] := b[n, i, k] = If[nJean-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).

A022816 Number of terms in 6th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 11, 44, 121, 271, 532, 952, 1590, 2517, 3817, 5588, 7943, 11011, 14938, 19888, 26044, 33609, 42807, 53884, 67109, 82775, 101200, 122728, 147730, 176605, 209781, 247716, 290899, 339851, 395126, 457312, 527032, 604945, 691747

Offset: 1

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A008778, A022811-A022818, A024207-A024210.

[n*(n+1)*(n^3+24*n^2+81*n-46)/120: n in [1..40]]; // Vincenzo Librandi, Oct 10 2011

Table[n(n+1)(n^3+24n^2+81n-46)/120,{n,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,44,121,271,532},40] (* Harvey P. Dale, Dec 29 2017 *)

a(n)=n*(n+1)*(n^3+24*n^2+81*n-46)/120 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = n*(n+1)*(n^3+24*n^2+81*n-46)/120. G.f.: x*(1+5*x-7*x^2+2*x^3)/(x-1)^6. - R. J. Mathar, Sep 15 2009

More terms from Christian G. Bower, Aug 15 1999.

A050301 Matrix 7th power of partition triangle A008284.

Original entry on oeis.org

1, 7, 1, 28, 7, 1, 105, 35, 7, 1, 322, 133, 35, 7, 1, 952, 455, 140, 35, 7, 1, 2541, 1379, 483, 140, 35, 7, 1, 6539, 3920, 1512, 490, 140, 35, 7, 1, 15833, 10375, 4354, 1540, 490, 140, 35, 7, 1, 37148, 26243, 11803, 4487, 1547, 490, 140, 35, 7, 1, 83594

Offset: 1

Christian G. Bower, Aug 15 1999

			1; 7,1; 28,7,1; 105,35,7,1; ...

Cf. A038497, A038498, A039805-A039807. A050300-A050304. a(n, 1) = A024207(n) (first column).

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A024209 Number of terms in n-th derivative of a function composed with itself 9 times.

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A022816 Number of terms in 6th derivative of a function composed with itself n times.

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A050301 Matrix 7th power of partition triangle A008284.

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