A011784 -id:A011784 - cp's OEIS frontend

A325282 Maximum adjusted frequency depth among integer partitions of n.

0, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (A325280).
Run lengths are A325258, i.e., first differences of Levine's sequence A011784 (except at n = 1).

Cf. A011784, A032741, A127002, A181819, A225486, A275870, A323014, A323023, A325245, A325254, A325258, A325278, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Max@@fdadj/@IntegerPartitions[n],{n,0,30}]

a(0) = 0; a(1) = 1; a(n > 1) = A225486(n).

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 3, 1, 3, 7, 10, 17, 27, 38, 1, 4, 8, 17, 31, 52, 83, 122, 181, 257, 361, 499, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 1, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754, 2570, 3742, 5269

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The Heinz numbers of these partitions are given by A325283.
The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014. The maximum adjusted frequency depth for integer partitions of n is given by A325282.
Essentially, the last numbers of rows of the array in A225485. - Clark Kimberling, Sep 13 2022

			The a(1) = 1 through a(11) = 17 partitions:
  1  11  21  211  221   411    3211  3221   3321    5221     4322
                  311   3111         4211   4221    5311     4331
                  2111  21111        32111  4311    6211     4421
                                            5211    32221    5411
                                            32211   33211    6221
                                            42111   42211    6311
                                            321111  43111    7211
                                                    52111    33221
                                                    421111   42221
                                                    3211111  43211
                                                             52211
                                                             53111
                                                             62111
                                                             431111
                                                             521111
                                                             4211111
                                                             32111111

Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

nn=30;
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==mfds[[n]]&]],{n,0,nn}]

A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

Original entry on oeis.org

2, 4, 6, 12, 18, 20, 24, 28, 40, 48, 60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280

Offset: 1

Gus Wiseman, Apr 17 2019

The enumeration of these partitions by sum is given by A325254.
The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
  2:   {1}         (1)
  4:   {1,1}       (2,1)
  6:   {1,2}       (2,2,1)
  12:  {1,1,2}     (3,2,2,1)
  18:  {1,2,2}     (3,2,2,1)
  20:  {1,1,3}     (3,2,2,1)
  24:  {1,1,1,2}   (4,2,2,1)
  28:  {1,1,4}     (3,2,2,1)
  40:  {1,1,1,3}   (4,2,2,1)
  48:  {1,1,1,1,2} (5,2,2,1)
  60:  {1,1,2,3}   (4,3,2,2,1)
  84:  {1,1,2,4}   (4,3,2,2,1)
  90:  {1,2,2,3}   (4,3,2,2,1)
  120: {1,1,1,2,3} (5,3,2,2,1)
  126: {1,2,2,4}   (4,3,2,2,1)
  132: {1,1,2,5}   (4,3,2,2,1)
  140: {1,1,3,4}   (4,3,2,2,1)
  150: {1,2,3,3}   (4,3,2,2,1)
  156: {1,1,2,6}   (4,3,2,2,1)
  168: {1,1,1,2,4} (5,3,2,2,1)
  180: {1,1,2,2,3} (5,3,2,2,1)

Cf. A011784, A056239, A112798, A118914, A181819, A182857, A225486, A323014, A323023, A325254, A325258, A325277, A325278, A325282.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

nn=30;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Select[Range[Prime[nn]],fdadj[primeMS[#]]==mfds[[Total[primeMS[#]]]]&]

A014622 Row sums of A014621.

Original entry on oeis.org

1, 1, 4, 29, 309, 4383, 78121, 1684706, 42801222, 1255919755, 41918624013, 1572257236114, 65619165625383, 3022617826829288, 152615633802149416, 8397224009015443509, 500957609480739613321, 32261529179806961067634, 2234133327582388824135291, 165800616088783604834296832

Offset: 1

Colin Mallows

nonn

Arises from an analysis of Levine's sequence A011784.

Max Alekseyev, Table of n, a(n) for n = 1..50

Cf. A011784, A014621.

More precise definition from R. J. Mathar, Sep 07 2008

Offset corrected and more terms added by Max Alekseyev, Sep 19 2023

A014644 Form array starting with {1,1}; then i-th term in a row gives number of i's in next row; sequence is formed from final term in each row.

Original entry on oeis.org

1, 2, 2, 3, 5, 11, 38, 272, 6474, 1090483, 4363282578, 2940715000315189, 7930047000157075949085439, 14412592242471457956514645440241289655074, 70636608026754077888330819116433040562582634705380432362008848092

Offset: 1

N. J. A. Sloane

			a(5)=5 because 5 is the last number of the 5th row of A014643, (1,2,2,3,3,4,4,4,5,5,5).

Cf. A014643, A011784.

NestList[Flatten@ MapIndexed[ConstantArray[First[#2], #1] &, #] &, {1, 1}, 8][[All, -1]] (* Michael De Vlieger, Dec 16 2021 *)

log a(n) grows like a constant times phi^n, where phi = golden ratio. - Colin Mallows
a(n) converges to a(n-2)*a(n-1)*phi (within 6 decimals for a(15)). - Johan Claes, Oct 02 2005
Limit_{n -> oo} a(n+2)/(a(n+1)*a(n)) = 1/phi. - Benoit Cloitre, Oct 13 2005

a(1)-a(11) computed by Colin Mallows

a(12)-a(15) computed by Johan Claes Oct 02 2005

A182858 Irregular table: Row n is the list of positive exponents in the canonical prime factorization of A182857(n), sorted in descending order, or 0 if there are no such exponents.

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 7, 6, 5, 4, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 14, 13, 12, 11, 10, 10, 9, 9, 8, 8, 7, 7, 7, 6, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 42, 41, 40, 39, 38, 37, 36, 35, 35, 34, 34, 33, 33, 32, 32, 31, 31

Offset: 0

Matthew Vandermast, Jan 05 2011

Length of row 0 = length of row 1 = 1; for n > 1, length of row n = A011784(n-1).

For n > 2, row n = row (n-2) of A012257, reversed.

			Table begins:
0,
1,
2,
1,1,
2,1,
2,1,1,
3,2,1,1,...

Cf. A181819, A182850.

A014643 Triangular array starting with {1,1}; then i-th term in a row gives number of i's in next row.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11

Offset: 1

N. J. A. Sloane

The row {2} could be safely prepended to this triangle. - Gus Wiseman, May 13 2018

			Triangle begins:
{1,1},
{1,2},
{1,2,2},
{1,2,2,3,3},
...

Alois P. Heinz, Rows n = 1..8, flattened

Rows converge to A001462.

Cf. A011784, A012257, A014643, A014644, A112798, A181819, A182850-A182858, A296150, A304455.

T:= proc(n) option remember; `if`(n=0, 2, (l->
      seq(i$l[i], i=1..nops(l)))([T(n-1)]))
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, May 17 2018

NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},8] (* Gus Wiseman, May 13 2018 *)

More terms from Patrick De Geest

A369988 Decimal expansion of Mallows's constant or stribolic constant kappa (of order 1).

Original entry on oeis.org

2, 7, 8, 8, 7, 7, 0, 6, 1

Offset: 0

Roland Miyamoto, Feb 07 2024

This constant is the area under the unique bijective, differentiable function h:[0,1]->[0,1] satisfying -c*h' = h^{-1} (compositional inverse) for some c > 0. That is, kappa = Integral_{t=0..1} h(t) dt, and then we also have kappa = c = -1/h'(0).
Equivalently, 1/kappa = 3.5858... is the only a > 0 such that there exists a differentiable function g:[0,a]->[0,a] which becomes its own derivative when rotated 90 degrees clockwise about the origin (into the fourth quadrant; whence the names "stribola" for g and h and "stribolic constant" for kappa, from Greek stribo=turn/twist), namely g(x):=h(kappa*x)/kappa for 0 <= x <= a = 1/kappa.
In 1997, Colin Mallows observed and conjectured that the rows in Levine's triangle A012257 take on stribolic shape and that A011784(n+1)/(A011784(n)*A011784(n-1)) converges as n->oo. Presuming his conjecture, the limit would equal kappa, while Mallows estimated it to be "approximately ... 0.277", see A011784. Later, in 2006, Martin Fuller suggested 0.27887706... for the limit, based on a numerical iteration, see A012257.
Set kappa_n := A369990(n) / A369991(n) and theta_n := (kappa_n-kappa_{n+1}) / (kappa_{n-1}-kappa_n). Under the hypothesis that theta_{2m} < theta_{2m+2} < theta_{2*m+3} < theta_{2*m+1} for m=1,2,... (verified for all values known so far), we would obtain 0.27887706136895087 < kappa_{21}' < kappa < kappa_{22}' < 0.27887706136898083, which is sharper than formula (3) below. Here, the transformed sequence (kappa_n') = G(kappa_n) is defined via kappa_n' := (kappa_{n-1}*kappa_{n+1} - kappa_n^2) / (kappa_{n-1} - 2*kappa_n + kappa_{n+1}). (See first arXiv article for a proof of this conjecture-dependent statement.) Feeling even more adventurous, we could apply the transformation G four times and would obtain 0.278877061368975064775 < kappa_{19}'''' < kappa < kappa_{18}'''' < 0.278877061368975064815.
It is an open question whether kappa is rational or irrational, algebraic or transcendental.

			0.278877061...

N. J. A. Sloane, My Favorite Integer Sequences, in: C. Ding, T. Helleseth, H. Niederreiter (editors), Sequences and their Applications, Discrete Mathematics and Theoretical Computer Science, Springer, London (1999) 103-130.

Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^{-1}, arXiv:2402.06618 [math.CO], 2024.
Roland Miyamoto, Solution to the iterative differential equation -gamma*g' = g^{-1}, arXiv:2404.11455 [math.CA], 2024.
Roland Miyamoto and J. W. Sander, Solving the iterative differential equation -gamma*g' = g^{-1}, in: H. Maier, J. & R. Steuding (eds.), Number Theory in Memory of Eduard Wirsing, Springer, 2023, pp. 223-236.
N. J. A. Sloane, My favorite integer sequences, in: Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Colin Mallows, and Bjorn Poonen, Discussion of A011784. [Scans of pages 150-155 and 164 of Sloane's notebook "Lattices 77", from June-July 1997.]

Cf. A369990, A369991, A369992, A369993, A011784, A012257, A014621.

Set kappa_n := A369990(n) / A369991(n).  Then
(1) kappa = lim_{n->oo}kappa_n = inf{kappa_n: n >= 0},
(2) kappa_n - 1 + kappa_n/kappa_{n-1} < kappa < kappa_n for n=1,2,...,
(3) 0.2788770612338 < kappa_{23} - 1 + kappa_{23}/kappa_{22} < kappa < kappa_{23} < 0.2788770613941.

A061892 Lionel-Levine-sequence generated by (1,0,0).

Original entry on oeis.org

0, 3, 1, 3, 3, 6, 10, 28, 108, 1011, 32511, 9314238, 84560776390, 219625370880235960

Offset: 0

Frank Ellermann, May 13 2001

A011784 allows the addition of a(-1)=0 and a(0)=2 using row A012257(-2)=1 and A012257(-1)=0 resp. row A012257(0)=2. In this sense A011784 / A012257 are "generated" by (1,0), A061892 / A061893 by (1,0,0), A061894 / A061895 by (2,0).

			a(4) = 3: (1,0,0), (3), (1,1,1), (1,2,3), (1,1,1,2,2,3).

Cf. A011784, A012257, A061893, A061894.

/* replace 1 0 0 with 1 0 to get A011784 */
S = ''; A = 1 0 0; do N = 1 to 10; T = words( A );
S = S word( A, T ); B = A; A = ''; do K = 1 to T;
A = A space( copies( K '', word( B, T + 1 - K )));
end K; end N; T = words( A ); say S word( A, T ) T;

a(11) from Naohiro Nomoto, May 10 2002
a(12) from Michael Anthony Keyes, Apr 14 2021
a(13) from Michael Anthony Keyes, May 13 2021

A144006 Triangle, read by rows of coefficients of x^ny^k for k=0..n(n-1)/2 for n>=0, defined by e.g.f.: A(x,y) = 1 + Series_Reversion( Integral A(-xy,y) dx ), with leading zeros in each row suppressed.

Original entry on oeis.org

1, 1, 1, 3, -1, 15, -10, 3, -1, 105, -105, 55, -30, 10, -3, 1, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1, 135135, -270270, 294525, -275275, 228375

Offset: 0

Paul D. Hanna, Sep 10 2008

Comment from Lucas Larsen, Aug 20 2024: (Start)
The nonzero entries in the n-th row appear to be the nonzero coefficients (up to sign) in the following:
 Let c be a fixed point in (0,oo) and f a smooth function such that f(c) = c and f(f'(x)) = x in a neighborhood of c. Then the n-th derivative of f evaluated at c can be written as a Laurent polynomial in c with the (descending) coefficients in question.
For instance:
 f'(c) = c
 f''(c) = c^(-1)
 f'''(c) = -c^(-4)
 f''''(c) = 3c^(-7) + c^(-8)
(End)

			Triangle begins (without suppressing leading zeros):
1;
1;
0, 1;
0,0, 3, -1;
0,0,0, 15, -10, 3, -1;
0,0,0,0, 105, -105, 55, -30, 10, -3, 1;
0,0,0,0,0, 945, -1260, 910, -630, 350, -168, 76, -30, 10, -3, 1;
0,0,0,0,0,0, 10395, -17325, 15750, -12880, 9135, -5789, 3381, -1806, 910, -434, 196, -76, 30, -10, 3, -1;
0,0,0,0,0,0,0, 135135, -270270, 294525, -275275, 228375, -172200, 120960, -78519, 48006, -28336, 16065, -8609, 4461, -2166, 1018, -470, 196, -76, 30, -10, 3, -1; ...

Cf. A144005, A067146.

Generates A014621, A014622 and A014623, which are related to Levine's sequence A011784.

{T(n,k)=local(A=1+x*O(x^n)); for(i=0,n,A=1+serreverse(intformal(subst(A,x,-x*y))));n!*polcoeff(polcoeff(A,n,x),k,y)}

#This is only correct if the observation in the comment from 2024/08/20 is true.
def T(n,k):
    if 0 <= n <= 1:
        return 1 if k == 0 else 0
    c = {(-1,):1} #Polynomial in infinitely many variables (function iterates)
    for _ in range(n-1):
        cnext = {}
        for key, value in c.items():
            key += (0,)
            for i, ni in enumerate(key):
                term = tuple(nj-2 if j==i else nj-1 if j<=i+1 else nj
                             for j,nj in enumerate(key))
                cnext[term] = cnext.get(term,0) + value*ni
                if cnext[term] == 0:
                    del cnext[term]
        c = cnext
    pairs = {} #Reduction to single variable (evaluation at fixpoint)
    for key, value in c.items():
        s = -sum(key)
        pairs[s] = pairs.get(s,0) + value
    _, row = zip(*sorted(pairs.items())) #Coefficients
    if 0 <= k-n+1 < len(row): #Correcting number of leading 0s
        return (-1)**(n+k+1)*abs(row[k-n+1]) #Correcting signs
    else:
        return 0
# Lucas Larsen, Aug 22 2024

E.g.f. satisfies: A(x,y) = 1 + Series_Reversion[Integral A(-x*y,y) dx].
T(n,k) = [x^n*y^k] n!*A(x,y) for k=0..n(n-1)/2, n>=0.
Row sums equal A144005.
A067146(n) = Sum_{k=0..n(n-1)/2} (-1)^k*T(n,k).
This is a signed version of table A014621 because setting f((1+x)/y):=A(-x*y,y)/y for fixed y>0 implies f(f(x))*f'(x)=-1 and f(1/y)=1/y, as in the second formula of A014621. Therefore, the row sums form A014623 and the unsigned row sums form A014622. - Roland Miyamoto, Jun 03 2024

cp's OEIS Frontend

A325282 Maximum adjusted frequency depth among integer partitions of n.

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A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

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A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

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A014622 Row sums of A014621.

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A014644 Form array starting with {1,1}; then i-th term in a row gives number of i's in next row; sequence is formed from final term in each row.

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A182858 Irregular table: Row n is the list of positive exponents in the canonical prime factorization of A182857(n), sorted in descending order, or 0 if there are no such exponents.

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A014643 Triangular array starting with {1,1}; then i-th term in a row gives number of i's in next row.

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A369988 Decimal expansion of Mallows's constant or stribolic constant kappa (of order 1).

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A061892 Lionel-Levine-sequence generated by (1,0,0).

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A144006 Triangle, read by rows of coefficients of x^ny^k for k=0..n(n-1)/2 for n>=0, defined by e.g.f.: A(x,y) = 1 + Series_Reversion( Integral A(-xy,y) dx ), with leading zeros in each row suppressed.