keyword:eigen - cp's OEIS frontend

A000748 Expansion of bracket function.

1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729, -2187, 4374, -6561, 6561, 0, -19683, 59049, -118098, 177147, -177147, 0, 531441, -1594323, 3188646, -4782969, 4782969, 0, -14348907, 43046721, -86093442, 129140163, -129140163, 0, 387420489, -1162261467

Offset: 0

N. J. A. Sloane

It appears that the sequence coincides with its third-order absolute difference. - John W. Layman, Sep 05 2003

It appears that, for n > 0, the (unsigned) a(n) = 3*|A057682(n)| = 3*|Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1)|. - John W. Layman, Sep 05 2003

			G.f. = 1 - 3*x + 6*x^2 - 9*x^3 + 9*x^4 - 27*x^6 + 81*x^7 - 162*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (-3,-3).

Column 3 of A307047.
Cf. A000749, A000750, A001659.
Cf. A057682.

I:=[1,-3]; [n le 2 select I[n] else -3*Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 11 2016

A000748:=(-1-2*z-3*z**2-3*z**3+18*z**5)/(-1+z+9*z**5); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from signs
a:= n-> (Matrix([[ -3,1], [ -3,0]])^n)[1,1]: seq(a(n), n=0..40); # Alois P. Heinz, Sep 06 2008

a[n_] := 2*3^(n/2)*Sin[(1-5*n)*Pi/6]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 12 2014 *)
LinearRecurrence[{-3, -3}, {1, -3}, 40] (* Jean-François Alcover, Feb 11 2016 *)

{a(n) = if( n<0, 0, polcoeff(1 / (1 + 3*x + 3*x^2) + x * O(x^n), n))}; /* Michael Somos, Jun 07 2005 */

{a(n) = if( n<0, 0, 3^((n+1)\2) * (-1)^(n\6) * ((-1)^n + (n%3==2)))}; /* Michael Somos, Sep 29 2007 */

G.f.: 1/((1+x)^3-x^3).
a(n) = A007653(3^n).
a(n) = -3*a(n-1) - 3*a(n-2). - Paul Curtz, May 12 2008
a(n) = Sum_{k=1..n} binomial(k,n-k)*(-3)^(k) for n > 0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
G.f.: 1/(1 + 3*x /(1 - x /(1+x))). - Michael Somos, May 12 2012
G.f.: G(0)/2, where G(k) = 1 + 1/( 1 - 3*x*(2*k+1 + x)/(3*x*(2*k+2 + x) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 09 2014
a(n) = 2*3^(n/2)*sin((1-5*n)*Pi/6). - Jean-François Alcover, Mar 12 2014
a(n) = (-1)^n * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+2,3*k+2). - Seiichi Manyama, Aug 05 2024
a(n) = (i*sqrt(3)/3)*((-3/2 - i*sqrt(3)/2)^(n+1) - (-3/2 + i*sqrt(3)/2)^(n+1)), where i = sqrt(-1). - Taras Goy, Jan 20 2025
a(n) = -2*a(n-1) + 3*a(n-3). - Taras Goy, Jan 26 2025

A035082 Number of rooted polygonal cacti (Husimi graphs) with n nodes.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 5, 13, 27, 67, 157, 390, 963, 2437, 6186, 15908, 41127, 107148, 280569, 738675, 1953054, 5185364, 13816018, 36934431, 99030038, 266254593, 717652816, 1938831589, 5249221790, 14240130827, 38702218134, 105367669062

Offset: 0

Christian G. Bower, Nov 15 1998

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71

Andrew Howroyd, Table of n, a(n) for n = 0..500
C. G. Bower, Transforms (2)
F. Harary and R. Z. Norman, The Dissimilarity Characteristic of Husimi Trees, Annals of Mathematics, 58 1953, pp. 134-141.
F. Harary and G. E. Uhlenbeck, On the Number of Husimi Trees, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953
N. J. A. Sloane, Transforms
Index entries for sequences related to cacti
Index entries for sequences related to rooted trees

Cf. A003080, A035083, A035084, A035085, A035086, A035087, A035088.

BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); concat([0], Vec(p))} \\ Andrew Howroyd, Aug 30 2018

Shifts left under transform T where Ta = EULER(BIK(a)-a).

A052318 Number of labeled rooted trimmed trees with n nodes.

Original entry on oeis.org

1, 2, 3, 16, 145, 1536, 19579, 290816, 4942305, 94689280, 2020278931, 47523053568, 1222147737265, 34117226135552, 1027550555918475, 33213871550365696, 1146891651823112641, 42135941698113503232, 1641164216596258397347, 67550839668807638712320

Offset: 1

Christian G. Bower, Dec 15 1999

A rooted trimmed tree is a tree with a forbidden limb of length 2.

A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Alois P. Heinz, Table of n, a(n) for n = 1..400
Index entries for sequences related to rooted trees

Cf. A002955, A002988-A002992, A052319-A052329.

A:= proc(n) option remember; if n<=1 then x else convert(series(x* exp(A(n-1)-x^2), x,n), polynom) fi end: a:= n-> coeff(A(n+1), x,n)*n!: seq(a(n), n=1..25); # Alois P. Heinz, Aug 23 2008

a[n_] := Sum[ Boole[ EvenQ[n-m]]*(m^((n+m)/2-2)/((n-m)/2)!)*((-1)^((n-m)/2)/(m-1)!), {m, 1, n}]*n!; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
Rest[CoefficientList[Series[-LambertW[-x/E^(x^2)],{x,0,20}],x]*Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum((if mod(n-m,2)=0 then m^((n+m)/2-2)/((n-m)/2)!*(-1)^((n-m)/2) else 0)/(m-1)!,m,1,n); /* Vladimir Kruchinin, Aug 07 2012 */

E.g.f. satisfies A(x) = x*exp(A(x) - x^2).
E.g.f.: -LambertW(-x/exp(x^2)). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ sqrt(1 + LambertW(-2*exp(-2))) * 2^(n/2) * n^(n-1) / (exp(n) * (-LambertW(-2*exp(-2)))^(n/2)). - Vaclav Kotesovec, Jan 08 2014

A330060 MM-numbers of VDD-normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 70, 72, 74, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128

Offset: 1

Gus Wiseman, Dec 03 2019

First differs from A330104 and A330120 in having 35 and lacking 69, with corresponding multisets of multisets 35: {{2},{1,1}} and 69: {{1},{2,2}}.
First differs from A330108 in having 207 and lacking 175, with corresponding multisets of multisets 207: {{1},{1},{2,2}} and 175: {{2},{2},{1,1}}.
We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all VDD-normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        49: {11}{11}         84: {}{}{1}{11}
   2: {}          24: {}{}{}{1}      52: {}{}{12}         89: {1112}
   3: {1}         26: {}{12}         53: {1111}           90: {}{1}{1}{2}
   4: {}{}        27: {1}{1}{1}      54: {}{1}{1}{1}      91: {11}{12}
   6: {}{1}       28: {}{}{11}       56: {}{}{}{11}       95: {2}{111}
   7: {11}        30: {}{1}{2}       57: {1}{111}         96: {}{}{}{}{}{1}
   8: {}{}{}      32: {}{}{}{}{}     60: {}{}{1}{2}       98: {}{11}{11}
   9: {1}{1}      35: {2}{11}        63: {1}{1}{11}      104: {}{}{}{12}
  12: {}{}{1}     36: {}{}{1}{1}     64: {}{}{}{}{}{}    105: {1}{2}{11}
  13: {12}        37: {112}          70: {}{2}{11}       106: {}{1111}
  14: {}{11}      38: {}{111}        72: {}{}{}{1}{1}    108: {}{}{1}{1}{1}
  15: {1}{2}      39: {1}{12}        74: {}{112}         111: {1}{112}
  16: {}{}{}{}    42: {}{1}{11}      76: {}{}{111}       112: {}{}{}{}{11}
  18: {}{1}{1}    45: {1}{1}{2}      78: {}{1}{12}       113: {123}
  19: {111}       48: {}{}{}{}{1}    81: {1}{1}{1}{1}    114: {}{1}{111}

Equals the image/fixed points of the idempotent sequence A330061.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Select[Range[100],Sort[primeMS/@primeMS[#]]==sysnorm[primeMS/@primeMS[#]]&]

A330099 BII-numbers of brute-force normalized set-systems.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 15, 19, 20, 21, 23, 31, 33, 37, 51, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 79, 83, 84, 85, 87, 95, 97, 101, 115, 116, 117, 119, 127, 139, 143, 159, 191, 203, 207, 223, 255, 267, 271, 275, 276, 277, 279, 287, 307, 308, 309, 311, 319, 331

Offset: 1

Gus Wiseman, Dec 02 2019

First differs from A330100 in having 545 and lacking 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the brute-force normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
There are A055621(n) entries m such that A326702(m) = n, where A326702(k) is the number of covered vertices in the set-system with BII-number k.
There are A283877(n) entries m such that A326031(m) = n, where A326031(k) is the weight of the set-system with BII-number k.

			The sequence of all nonempty brute-force normalized set-systems together with their BII-numbers begins:
   1: {1}                  52: {12}{13}{23}
   3: {1}{2}               53: {1}{12}{13}{23}
   4: {12}                 55: {1}{2}{12}{13}{23}
   5: {1}{12}              63: {1}{2}{3}{12}{13}{23}
   7: {1}{2}{12}           64: {123}
  11: {1}{2}{3}            65: {1}{123}
  15: {1}{2}{3}{12}        67: {1}{2}{123}
  19: {1}{2}{13}           68: {12}{123}
  20: {12}{13}             69: {1}{12}{123}
  21: {1}{12}{13}          71: {1}{2}{12}{123}
  23: {1}{2}{12}{13}       75: {1}{2}{3}{123}
  31: {1}{2}{3}{12}{13}    79: {1}{2}{3}{12}{123}
  33: {1}{23}              83: {1}{2}{13}{123}
  37: {1}{12}{23}          84: {12}{13}{123}
  51: {1}{2}{13}{23}       85: {1}{12}{13}{123}

Equals the image/fixed points of the idempotent sequence A330101.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems by span are A055621.
Unlabeled spanning set-systems by weight are A283877.
Cf. A000612, A300913, A321405, A330061, A330102, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[0,100],Sort[bpe/@bpe[#]]==brute[bpe/@bpe[#]]&]

A330100 BII-numbers of VDD-normalized set-systems.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 15, 19, 20, 21, 23, 31, 33, 37, 51, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 79, 83, 84, 85, 87, 95, 97, 101, 115, 116, 117, 119, 127, 139, 143, 159, 179, 180, 181, 183, 191, 203, 207, 211, 212, 213, 215, 223, 225, 229, 243, 244, 245, 247

Offset: 0

Gus Wiseman, Dec 04 2019

First differs from A330099 in lacking 545 and having 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}

			The sequence of all nonempty VDD-normalized set-systems together with their BII-numbers begins:
   1: {1}                  52: {12}{13}{23}
   3: {1}{2}               53: {1}{12}{13}{23}
   4: {12}                 55: {1}{2}{12}{13}{23}
   5: {1}{12}              63: {1}{2}{3}{12}{13}{23}
   7: {1}{2}{12}           64: {123}
  11: {1}{2}{3}            65: {1}{123}
  15: {1}{2}{3}{12}        67: {1}{2}{123}
  19: {1}{2}{13}           68: {12}{123}
  20: {12}{13}             69: {1}{12}{123}
  21: {1}{12}{13}          71: {1}{2}{12}{123}
  23: {1}{2}{12}{13}       75: {1}{2}{3}{123}
  31: {1}{2}{3}{12}{13}    79: {1}{2}{3}{12}{123}
  33: {1}{23}              83: {1}{2}{13}{123}
  37: {1}{12}{23}          84: {12}{13}{123}
  51: {1}{2}{13}{23}       85: {1}{12}{13}{123}

Equals the image/fixed points of the idempotent sequence A330102.
A subset of A326754.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A330061, A330101.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Select[Range[0,100],Sort[bpe/@bpe[#]]==sysnorm[bpe/@bpe[#]]&]

A330104 MM-numbers of brute-force normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 69, 72, 74, 76, 78, 81, 84, 89, 90, 91, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128, 131, 133

Offset: 1

Gus Wiseman, Dec 02 2019

First differs from A330060 and A330108 in having 69 and lacking 35, with corresponding multisets of multisets 69: {{1},{2,2}} and 35: {{2},{1,1}}.
First differs from A330120 in having 435 and lacking 429, with corresponding multisets of multisets 435: {{1},{2},{1,3}} and 429: {{1},{3},{1,2}}.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all brute-force normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        52: {}{}{12}         89: {1112}
   2: {}          24: {}{}{}{1}      53: {1111}           90: {}{1}{1}{2}
   3: {1}         26: {}{12}         54: {}{1}{1}{1}      91: {11}{12}
   4: {}{}        27: {1}{1}{1}      56: {}{}{}{11}       96: {}{}{}{}{}{1}
   6: {}{1}       28: {}{}{11}       57: {1}{111}         98: {}{11}{11}
   7: {11}        30: {}{1}{2}       60: {}{}{1}{2}      104: {}{}{}{12}
   8: {}{}{}      32: {}{}{}{}{}     63: {1}{1}{11}      105: {1}{2}{11}
   9: {1}{1}      36: {}{}{1}{1}     64: {}{}{}{}{}{}    106: {}{1111}
  12: {}{}{1}     37: {112}          69: {1}{22}         108: {}{}{1}{1}{1}
  13: {12}        38: {}{111}        72: {}{}{}{1}{1}    111: {1}{112}
  14: {}{11}      39: {1}{12}        74: {}{112}         112: {}{}{}{}{11}
  15: {1}{2}      42: {}{1}{11}      76: {}{}{111}       113: {123}
  16: {}{}{}{}    45: {1}{1}{2}      78: {}{1}{12}       114: {}{1}{111}
  18: {}{1}{1}    48: {}{}{}{}{1}    81: {1}{1}{1}{1}    117: {1}{1}{12}
  19: {111}       49: {11}{11}       84: {}{}{1}{11}     120: {}{}{}{1}{2}

Equals the image/fixed points of the idempotent sequence A330105.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330101, A330103.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[100],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A330107 MM-numbers of brute-force normalized multiset partitions.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 37, 39, 45, 49, 53, 57, 63, 69, 81, 89, 91, 105, 111, 113, 117, 131, 133, 135, 141, 147, 151, 159, 161, 165, 169, 171, 183, 189, 195, 207, 223, 225, 243, 247, 259, 267, 273, 281, 285, 309, 311, 315, 329, 333, 339, 343, 351, 359

Offset: 1

Gus Wiseman, Dec 02 2019

A multiset partition is a finite multiset of finite nonempty multisets of positive integers.
We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all brute-force normalized multiset partitions together with their MM-numbers begins:
   1: 0             63: {1}{1}{11}      159: {1}{1111}
   3: {1}           69: {1}{22}         161: {11}{22}
   7: {11}          81: {1}{1}{1}{1}    165: {1}{2}{3}
   9: {1}{1}        89: {1112}          169: {12}{12}
  13: {12}          91: {11}{12}        171: {1}{1}{111}
  15: {1}{2}       105: {1}{2}{11}      183: {1}{122}
  19: {111}        111: {1}{112}        189: {1}{1}{1}{11}
  21: {1}{11}      113: {123}           195: {1}{2}{12}
  27: {1}{1}{1}    117: {1}{1}{12}      207: {1}{1}{22}
  37: {112}        131: {11111}         223: {11112}
  39: {1}{12}      133: {11}{111}       225: {1}{1}{2}{2}
  45: {1}{1}{2}    135: {1}{1}{1}{2}    243: {1}{1}{1}{1}{1}
  49: {11}{11}     141: {1}{23}         247: {12}{111}
  53: {1111}       147: {1}{11}{11}     259: {11}{112}
  57: {1}{111}     151: {1122}          267: {1}{1112}

Equals the odd terms of A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A300300, A316983, A317533, A320664, A330061, A330098, A330101, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[1,100,2],Sort[primeMS/@primeMS[#]]==brute[primeMS/@primeMS[#]]&]

A330108 MM-numbers of MM-normalized multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 63, 64, 70, 72, 74, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117, 120, 126, 128

Offset: 1

Gus Wiseman, Dec 05 2019

First differs from A330060 in having 175 and lacking 207, with corresponding multisets of multisets 175: {{2},{2},{1,1}} and 207: {{1},{1},{2,2}}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
We define the MM-normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest MM-number.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}

			The sequence of all MM-normalized multisets of multisets together with their MM-numbers begins:
   1: 0           21: {1}{11}        49: {11}{11}         84: {}{}{1}{11}
   2: {}          24: {}{}{}{1}      52: {}{}{12}         89: {1112}
   3: {1}         26: {}{12}         53: {1111}           90: {}{1}{1}{2}
   4: {}{}        27: {1}{1}{1}      54: {}{1}{1}{1}      91: {11}{12}
   6: {}{1}       28: {}{}{11}       56: {}{}{}{11}       95: {2}{111}
   7: {11}        30: {}{1}{2}       57: {1}{111}         96: {}{}{}{}{}{1}
   8: {}{}{}      32: {}{}{}{}{}     60: {}{}{1}{2}       98: {}{11}{11}
   9: {1}{1}      35: {2}{11}        63: {1}{1}{11}      104: {}{}{}{12}
  12: {}{}{1}     36: {}{}{1}{1}     64: {}{}{}{}{}{}    105: {1}{2}{11}
  13: {12}        37: {112}          70: {}{2}{11}       106: {}{1111}
  14: {}{11}      38: {}{111}        72: {}{}{}{1}{1}    108: {}{}{1}{1}{1}
  15: {1}{2}      39: {1}{12}        74: {}{112}         111: {1}{112}
  16: {}{}{}{}    42: {}{1}{11}      76: {}{}{111}       112: {}{}{}{}{11}
  18: {}{1}{1}    45: {1}{1}{2}      78: {}{1}{12}       113: {123}
  19: {111}       48: {}{}{}{}{1}    81: {1}{1}{1}{1}    114: {}{1}{111}

Equals the image/fixed points of the idempotent sequence A330194.
A subset of A320456.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A056239, A112798, A317533, A330061, A330098, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Select[Range[100],Sort[primeMS/@primeMS[#]]==mmnorm[primeMS/@primeMS[#]]&]

A003659 Shifts left under Stirling2 transform.

Original entry on oeis.org

1, 1, 2, 6, 26, 152, 1144, 10742, 122772, 1673856, 26780972, 496090330, 10519217930, 252851833482, 6832018188414, 205985750827854, 6885220780488694, 253685194149119818, 10250343686634687424, 452108221967363310278, 21676762640915055856716

Offset: 1

N. J. A. Sloane, Mira Bernstein

Apart from leading term, number of M-sequences from multicomplexes on at most 4 variables with no monomial of degree more than n+1.
Stirling2 transform of a(n) = [1, 1, 2, 6, 26, ...] is a(n+1) = [1, 2, 6, 26, ...].
Eigensequence of Stirling2 triangle A008277. - Philippe Deléham, Mar 23 2007

S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..330
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. [_N. J. A. Sloane_, Sep 16 2012]
Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047 [math.CO], 2008. [_Jonathan Vos Post_, Dec 22 2008]
N. J. A. Sloane, Transforms

Cf. A048801.

Cf. A153277, A153278. - Jonathan Vos Post, Dec 22 2008

stirtr:= proc(p)
           proc(n) add(p(k)*Stirling2(n,k), k=0..n) end
         end:
a:= proc(n) option remember; `if`(n<3, 1, aa(n-1)) end:
aa:= stirtr(a):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 22 2012

terms = 21; A[] = 0; Do[A[x] = Normal[Integrate[1 + A[Exp[x] - 1 + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms];
CoefficientList[A[x], x]*Range[0, terms]! // Rest (* Jean-François Alcover, May 23 2012, updated Jan 12 2018 *)

{a(n)=local(A, E); if(n<0, 0, A=O(x); E=exp(x+x*O(x^n))-1; for(m=1, n, A=intformal( subst( 1+A, x, E+x*O(x^m)))); n!*polcoeff(A, n))} /* Michael Somos, Mar 08 2004 */

a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=sum(j=1, i, stirling(i, j, 2)*v[j])); v; \\ Seiichi Manyama, Jun 24 2022

E.g.f. A(x) satisfies A(x)' = 1+A(exp(x)-1).
G.f. satisfies: Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{n>=1} a(n) * x^n / Product_{j=1..n} (1 - j*x)). - Ilya Gutkovskiy, May 09 2019
a(1) = 1; a(n+1) = Sum_{k=1..n} Stirling2(n,k) * a(k). - Seiichi Manyama, Jun 24 2022

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