A055884
Euler transform of partition triangle A008284.
Original entry on oeis.org
1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 7, 7, 1, 6, 12, 16, 12, 11, 1, 6, 17, 25, 28, 19, 15, 1, 8, 22, 43, 49, 48, 30, 22, 1, 8, 30, 58, 87, 88, 77, 45, 30, 1, 10, 36, 87, 134, 167, 151, 122, 67, 42, 1, 10, 45, 113, 207, 270, 296, 247, 185, 97, 56, 1, 12, 54, 155, 295, 448, 510, 507, 394, 278, 139, 77
Offset: 1
From _Gus Wiseman_, Nov 09 2018: (Start)
Triangle begins:
1
1 2
1 2 3
1 4 4 5
1 4 8 7 7
1 6 12 16 12 11
1 6 17 25 28 19 15
1 8 22 43 49 48 30 22
1 8 30 58 87 88 77 45 30
...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
{{5}} {{1,4}} {{1,1,3}} {{1,1,1,2}} {{1,1,1,1,1}}
{{2,3}} {{1,2,2}} {{1},{1,1,2}} {{1},{1,1,1,1}}
{{1},{4}} {{1},{1,3}} {{1,1},{1,2}} {{1,1},{1,1,1}}
{{2},{3}} {{1},{2,2}} {{2},{1,1,1}} {{1},{1},{1,1,1}}
{{2},{1,2}} {{1},{1},{1,2}} {{1},{1,1},{1,1}}
{{3},{1,1}} {{1},{2},{1,1}} {{1},{1},{1},{1,1}}
{{1},{1},{3}} {{1},{1},{1},{2}} {{1},{1},{1},{1},{1}}
{{1},{2},{2}}
(End)
-
h:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n,k), k=1..n), n=1..12); # Alois P. Heinz, Feb 17 2023
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* Gus Wiseman, Nov 09 2018 *)
A007713
Number of 4-level rooted trees with n leaves.
Original entry on oeis.org
1, 1, 4, 10, 30, 75, 206, 518, 1344, 3357, 8429, 20759, 51044, 123973, 299848, 719197, 1716563, 4070800, 9607797, 22555988, 52718749, 122655485, 284207304, 655894527, 1508046031, 3454808143, 7887768997, 17949709753, 40719611684, 92096461012, 207697731344
Offset: 0
From _Gus Wiseman_, Oct 11 2018: (Start)
Also the number of multiset partitions of multiset partitions of integer partitions of n. For example, the a(1) = 1 through a(4) = 30 multiset partitions are:
((1)) ((2)) ((3)) ((4))
((11)) ((12)) ((13))
((1)(1)) ((111)) ((22))
((1))((1)) ((1)(2)) ((112))
((1)(11)) ((1111))
((1))((2)) ((1)(3))
((1))((11)) ((2)(2))
((1)(1)(1)) ((1)(12))
((1))((1)(1)) ((2)(11))
((1))((1))((1)) ((1)(111))
((11)(11))
((1))((3))
((2))((2))
((1))((12))
((1)(1)(2))
((2))((11))
((1))((111))
((1)(1)(11))
((11))((11))
((1))((1)(2))
((2))((1)(1))
((1))((1)(11))
((1)(1)(1)(1))
((11))((1)(1))
((1))((1))((2))
((1))((1))((11))
((1))((1)(1)(1))
((1)(1))((1)(1))
((1))((1))((1)(1))
((1))((1))((1))((1))
(End)
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
- B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
- N. J. A. Sloane, Transforms
- Index entries for sequences related to rooted trees
Cf.
A001970,
A047968,
A050342,
A089259,
A141268,
A258466,
A261049,
A319066,
A320328,
A320330,
A320331.
-
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # Alois P. Heinz, Sep 08 2008
-
i[ n_, m_ ] := 1 /; m==1 || n==0; i[ n_, m_ ] := (i[ n, m ]=1/n Sum[ i[ k, m ] Plus @@ ((# i[ #, m-1 ])& /@ Divisors[ n-k ]), {k, 0, n-1} ]) /; n>0 && m>1
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b0 = etr[Function[1]]; b1 = etr[b0]; a = etr[b1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)
A055885
Euler transform applied twice to partition triangle A008284.
Original entry on oeis.org
1, 1, 3, 1, 3, 6, 1, 6, 9, 14, 1, 6, 18, 23, 27, 1, 9, 27, 54, 57, 58, 1, 9, 39, 87, 140, 131, 111, 1, 12, 51, 150, 259, 353, 295, 223, 1, 12, 69, 210, 470, 702, 832, 637, 424, 1, 15, 84, 314, 749, 1379, 1803, 1917, 1350, 817, 1, 15, 105, 416, 1176, 2352, 3730, 4403, 4245, 2789, 1527
Offset: 1
1;
1, 3;
1, 3, 6;
1, 6, 9, 14;
1, 6, 18, 23, 27;
...
Showing 1-3 of 3 results.
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