A022818 -id:A022818 - cp's OEIS frontend

A024208 Number of terms in n-th derivative of a function composed with itself 8 times.

1, 1, 8, 36, 148, 498, 1590, 4586, 12644, 32775, 81901, 196085, 455772, 1025779, 2252674, 4823546, 10116553, 20783490, 41949270, 83211931, 162552093, 312850854, 594086542, 1113610526, 2062796698, 3777567977, 6844786250, 12276620372, 21809737429, 38391720375

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 A050302.

Column k=8 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).

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

Original entry on oeis.org

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.

A192435 Number of terms in n-th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 2, 6, 26, 110, 532, 2541, 12644, 63024, 318857, 1618947, 8277062, 42453073, 218597485, 1128527057, 5841301830, 30297014746, 157442596130, 819511659381, 4272054888643, 22299423992018, 116539878029773, 609718298887977, 3193136462042241, 16737951567806110

Offset: 1

Alois P. Heinz, Aug 18 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..600
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Main diagonal of A022818.

Cf. A008485.

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 A(n, n):
seq(a(n), n=1..40);

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[nJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.35270133348664..., c = 0.0504640078963302151598181537452... . - Vaclav Kotesovec, Sep 03 2014, updated May 19 2018

A215626 Number of terms in 8th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 22, 129, 468, 1309, 3101, 6539, 12644, 22857, 39148, 64141, 101256, 154869, 230491, 334967, 476696, 665873, 914754, 1237945, 1652716, 2179341, 2841465, 3666499, 4686044, 5936345, 7458776, 9300357, 11514304, 14160613, 17306679, 21027951, 25408624, 30542369

Offset: 1

Alois P. Heinz, Aug 18 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Row n=8 of A022818.

a:= n-> n*(n+6)*(92+(-713+(722+(568+(50+n)*n)*n)*n)*n)/5040:
seq(a(n), n=1..40);

CoefficientList[Series[(x^6-7x^5+15x^4-4x^3-19x^2+14x+1)x/(x-1)^8,{x,0,40}],x]//Rest (* Harvey P. Dale, Aug 02 2020 *)

G.f.: (x^6-7*x^5+15*x^4-4*x^3-19*x^2+14*x+1)*x/(x-1)^8.

a(n) = n*(n+6)*(n^5+50*n^4+568*n^3+722*n^2-713*n+92)/5040.

A215627 Number of terms in 9th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 30, 210, 867, 2690, 6956, 15833, 32775, 63024, 114235, 197241, 326976, 523575, 813671, 1231910, 1822706, 2642259, 3760860, 5265508, 7262865, 9882576, 13280982, 17645255, 23197985, 30202250, 38967201, 49854195, 63283510, 79741677, 99789465, 124070556

Offset: 1

Alois P. Heinz, Aug 18 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Row n=9 of A022818.

a:= n-> n*(n+1)*(6000+(-30788+(29472+(13697+(1703+(75+n)*n)*n)*n)*n)*n)/40320:
seq(a(n), n=1..40);

G.f.: (5*x^6-28*x^5+53*x^4-27*x^3-24*x^2+21*x+1)*x/(1-x)^9.

a(n) = n*(n+1)*(n^6+75*n^5+1703*n^4+13697*n^3+29472*n^2-30788*n+6000)/40320.

A215628 Number of terms in 10th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 42, 345, 1597, 5436, 15217, 37148, 81901, 166819, 318857, 578413, 1004224, 1679522, 2719666, 4281488, 6574614, 9875045, 14541308, 21033513, 29935679, 41981720, 58085511, 79375484, 107234235, 143343655, 189736131, 248852397, 323606650, 417459582, 534500016

Offset: 1

Alois P. Heinz, Aug 18 2012

Alois P. Heinz, Table of n, a(n) for n = 1..1000
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Row n=10 of A022818.

a:= n-> n*(n+1)*(136656+(-412380+(209204+(203887+(40826+(3148+(98+n)*
        n)*n)*n)*n)*n)*n)/362880:
seq(a(n), n=1..40);

CoefficientList[Series[-(4*x^7 - 34*x^6 + 110*x^5 - 161*x^4 + 83*x^3 + 30*x^2 - 32*x - 1)/(x - 1)^10, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 27 2017 *)

G.f.: -(4*x^7-34*x^6+110*x^5-161*x^4+83*x^3+30*x^2-32*x-1)*x/(x-1)^10.

a(n) = n*(n+1)*(n^7 +98*n^6 +3148*n^5 +40826*n^4 +203887*n^3 +209204*n^2 -412380*n +136656) / 362880.

cp's OEIS Frontend

A024208 Number of terms in n-th derivative of a function composed with itself 8 times.

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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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A192435 Number of terms in n-th derivative of a function composed with itself n times.

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A215626 Number of terms in 8th derivative of a function composed with itself n times.

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A215627 Number of terms in 9th derivative of a function composed with itself n times.

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A215628 Number of terms in 10th derivative of a function composed with itself n times.

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