A001383 -id:A001383 - cp's OEIS frontend

A301595 Number of thrice-partitions of n.

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508

Offset: 0

Gus Wiseman, Mar 24 2018

nonn

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019

twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
     1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A001384 Number of n-node trees of height at most 4.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 19, 42, 89, 191, 402, 847, 1763, 3667, 7564, 15564, 31851, 64987, 132031, 267471, 539949, 1087004, 2181796, 4367927, 8721533, 17372967, 34524291, 68456755, 135446896, 267444085, 527027186, 1036591718, 2035083599

Offset: 0

N. J. A. Sloane

nonn

a(n+1) is also the number of n-vertex graphs that do not contain a P_4, C_4, or K_5 as induced subgraph (K_5-free trivially perfect graphs, cf. A123467). - Falk Hüffner, Jan 10 2016

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).

N. J. A. Sloane, Table of n, a(n) for n=0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 63
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

See A001383 for details.

For Maple program see link in A000235.
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: A000041:= etr(n->1): b1:= etr(k-> A000041(k-1)): A001383:= n->`if`(n=0,1,b1(n-1)): b2:= etr(A001383): a:= n->`if`(n=0,1,b2(n-1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

Prepend[Nest[CoefficientList[Series[Product[1/(1-x^i)^#[[i]],{i,1,Length[#]}],{x,0,40}],x]&,{1},4],1] (* Geoffrey Critzer, Aug 01 2013 *)

Take Euler transform of A001383 and shift right. (Christian G. Bower)

A034824 Number of n-node rooted trees of height at most 7.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 285, 710, 1789, 4514, 11431, 28922, 73182, 184917, 466755, 1176393, 2961205, 7443770, 18689435, 46869152, 117412440, 293832126, 734645046, 1835147741, 4580420719, 11423511895, 28469058647, 70899220083, 176449174539, 438854372942

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Table of n, a(n) for n=0..200
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

See A001383 for details.

For Maple program see link in A000235.
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: shr:= proc(p) n->`if`(n=0, 1,p(n-1)) end: b[0]:= etr(n->1): for j from 1 to 5 do b[j]:= etr(shr(b[j-1])) od: a:= shr(b[5]): seq(a(n), n=0..35); # Alois P. Heinz, Sep 08 2008

Prepend[Nest[CoefficientList[Series[Product[1/(1-x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x]&, {1}, 7], 1] (* Geoffrey Critzer, Aug 01 2013 *)

Take Euler transform of A034823 and shift right. (Christian G. Bower).

A218551 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(nk)A(x^k)^n) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 106, 322, 987, 3119, 9985, 32499, 106910, 355524, 1191960, 4026739, 13689783, 46807685, 160842381, 555175377, 1923970425, 6691769948, 23351250882, 81729943060, 286842588316, 1009256119760, 3559337691300, 12579738946685, 44549347255523, 158058591860684

Offset: 0

Paul D. Hanna, Nov 01 2012

nonn

Compare to the dual g.f. G(x) of A219231:

G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*G(x^n)^k) ).

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 106*x^6 + 322*x^7 +...
where
log(A(x)) = x/(1*(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
x^2/(2*(1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
x^3/(3*(1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
x^4/(4*(1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 106*x^5/5 + 342*x^6/6 + 1198*x^7/7 + 4071*x^8/8 + 14356*x^9/9 + 50408*x^10/10 +...

Cf. A219231, A001383, A218552, A218575.

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m/m*prod(k=1,n\m+1,1/(1-x^(m*k)*subst(A,x,x^k +x*O(x^n))^m)))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A302094 Number of relatively prime or monic twice-partitions of n.

Original entry on oeis.org

1, 3, 6, 10, 27, 35, 113, 170, 396, 641, 1649, 2318, 5905, 9112, 18678, 32529, 69094, 106210, 227480, 363433, 705210, 1196190, 2325023, 3724233, 7192245, 11915884, 21857887, 36597843, 67406158, 109594872, 201747847, 333400746, 591125465, 987069077, 1743223350

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime). Then a relatively prime or monic twice-partition of n is a choice of a relatively prime or monic partition of each part in a relatively prime or monic partition of n.

			The a(4) = 10 relatively prime or monic twice-partitions:
(4), (31), (211), (1111),
(3)(1), (21)(1), (111)(1),
(2)(1)(1), (11)(1)(1),
(1)(1)(1)(1).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A300486, A301462, A301467, A301480, A302915, A302916, A302917.

ip[n_]:=ip[n]=Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&];
Table[Sum[Times@@Length/@ip/@ptn,{ptn,ip[n]}],{n,10}]

A034825 Number of n-node rooted trees of height at most 8.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 718, 1832, 4702, 12159, 31515, 81888, 212878, 553557, 1438741, 3737331, 9700188, 25156049, 65181067, 168746672, 436505846, 1128256918, 2914103577, 7521450053, 19400577711, 50010551503, 128841990772, 331754004302

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Table of n, a(n) for n=0..200
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

See A001383 for details.

For Maple program see link in A000235.
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: shr:= proc(p) n->`if`(n=0, 1,p(n-1)) end: b[0]:= etr(n->1): for j from 1 to 6 do b[j]:= etr(shr(b[j-1])) od: a:= shr(b[6]): seq(a(n), n=0..31); # Alois P. Heinz, Sep 08 2008

Prepend[Nest[CoefficientList[Series[Product[1/(1-x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x]&, {1}, 8], 1] (* Geoffrey Critzer, Aug 01 2013 *)

Take Euler transform of A034824 and shift right. (Christian G. Bower).

A000299 Number of n-node rooted trees of height 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 13, 36, 93, 225, 528, 1198, 2666, 5815, 12517, 26587, 55933, 116564, 241151, 495417, 1011950, 2055892, 4157514, 8371318, 16792066, 33564256, 66875221, 132849983, 263192599, 520087551, 1025295487, 2016745784, 3958608430, 7754810743

Offset: 1

N. J. A. Sloane

nonn

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 = 1..1000 (first 200 terms from N. J. A. Sloane)
F. Harary & R. W. Robinson, The number of achiral trees, Jnl. Reine Angewandte Mathematik 278 (1975), 322-335. (Annotated scanned copy)
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Column h=4 of A034781.

Maple
```
For Maple program see link in A000235.
```

f[n_] := Nest[CoefficientList[Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 40}], x] &, {1}, n];f[4]-f[3] (* Geoffrey Critzer, Aug 01 2013 *)

Cf. A001384-A001383. - Christian G. Bower

A034826 Number of n-node rooted trees of height at most 9.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1841, 4755, 12410, 32558, 85849, 226980, 601373, 1594870, 4232100, 11230771, 29798539, 79034638, 209526631, 555172356, 1470195001, 3891131705, 10292857772, 27212082536, 71905725130, 189911518888

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Table of n, a(n) for n=0..200
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

See A001383 for details.

For Maple program see link in A000235.
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: shr:= proc(p) n->`if`(n=0, 1,p(n-1)) end: b[0]:= etr(n->1): for j from 1 to 7 do b[j]:= etr(shr(b[j-1])) od: a:= shr(b[7]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

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]; shr[p_] = If[# == 0, 1, p[#-1]]&; b[0] = etr[1&]; For[j = 1, j <= 7, j++, b[j] = etr[shr[b[j-1]]]]; a = shr[b[7]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

Take Euler transform of A034825 and shift right. (Christian G. Bower).

cp's OEIS Frontend

A301595 Number of thrice-partitions of n.

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A001384 Number of n-node trees of height at most 4.

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A034824 Number of n-node rooted trees of height at most 7.

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A218551 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^k)^n) ).

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PARI

A302094 Number of relatively prime or monic twice-partitions of n.

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Mathematica

A034825 Number of n-node rooted trees of height at most 8.

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Maple

Mathematica

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A000299 Number of n-node rooted trees of height 4.

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Maple

Mathematica

Formula

A034826 Number of n-node rooted trees of height at most 9.

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A218551 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(nk)A(x^k)^n) ).