A290353 -id:A290353 - cp's OEIS frontend

A290356 The seventh Euler transform of the sequence with g.f. 1+x.

1, 1, 7, 28, 140, 602, 2772, 12166, 54046, 236093, 1030101, 4458247, 19223202, 82448782, 352247250, 1498724840, 6353940527, 26844401919, 113051495750, 474652297902, 1987159118837, 8296760311018, 34551340915438, 143533939056129, 594877730354756

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

Also the number of 7-level rooted trees with n leaves. All n leaves are in level 7.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees

Column k=7 of A290353.

Cf. A290355.

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
    end:
a:= n-> b(n, 7):
seq(a(n), n=0..30);

b[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[b[d, k - 1]*d, {d, Divisors[j]}] b[n - j, k], {j, n}]/n]]; Table[b[n, 7], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f.: Product_{j>0} 1/(1-x^j)^A290355(j).

A290357 The eighth Euler transform of the sequence with g.f. 1+x.

Original entry on oeis.org

1, 1, 8, 36, 204, 1002, 5244, 26328, 133476, 667335, 3331117, 16516607, 81607176, 401407499, 1967534543, 9609826869, 46788348316, 227114265649, 1099339308308, 5307155062783, 25556511343601, 122773840789344, 588473630650319, 2814565652799711, 13433897987956859

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

Also the number of 8-level rooted trees with n leaves. All n leaves are in level 8.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees

Column k=8 of A290353.

Cf. A290356.

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
    end:
a:= n-> b(n, 8):
seq(a(n), n=0..30);

b[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[b[d, k - 1]*d, {d, Divisors[j]}] b[n - j, k], {j, n}]/n]]; Table[b[n, 8], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f.: Product_{j>0} 1/(1-x^j)^A290356(j).

A290358 The ninth Euler transform of the sequence with g.f. 1+x.

Original entry on oeis.org

1, 1, 9, 45, 285, 1575, 9237, 52221, 297633, 1676364, 9425128, 52688379, 293582296, 1629482947, 9015732880, 49727160669, 273504111761, 1500271605182, 8209029290412, 44811239964075, 244069307558722, 1326536980923855, 7195340066129605, 38953817605037254

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

Also the number of 9-level rooted trees with n leaves. All n leaves are in level 9.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees

Column k=9 of A290353.

Cf. A290357.

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
    end:
a:= n-> b(n, 9):
seq(a(n), n=0..30);

b[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[b[d, k - 1]*d, {d, Divisors[j]}] b[n - j, k], {j, n}]/n]]; Table[b[n, 9], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f.: Product_{j>0} 1/(1-x^j)^A290357(j).

A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 4, 1, 6, 1, 1, 1, 1, 2, 6, 1, 5, 1, 7, 1, 1, 1, 1, 2, 3, 10, 1, 6, 1, 8, 1, 1, 1, 1, 1, 3, 4, 15, 1, 7, 1, 9, 1, 1, 1, 1, 4, 1, 4, 5, 21, 1, 8, 1, 10, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 25 2019

An orderless factorization of n with k > 1 levels of parentheses is any multiset partition of an orderless factorization of n with k - 1 levels of parentheses. If k = 1 it is just an orderless factorization of n into factors > 1.

			Array begins:
       k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12
   n=1: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=2: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=3: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=4: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=5: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=6: 1    2    3    4    5    6    7    8    9   10   11   12   13
   n=7: 1    1    1    1    1    1    1    1    1    1    1    1    1
   n=8: 1    3    6   10   15   21   28   36   45   55   66   78   91
   n=9: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=10: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=11: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=12: 1    4    9   16   25   36   49   64   81  100  121  144  169
  n=13: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=14: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=15: 1    2    3    4    5    6    7    8    9   10   11   12   13
  n=16: 1    5   14   30   55   91  140  204  285  385  506  650  819
  n=17: 1    1    1    1    1    1    1    1    1    1    1    1    1
  n=18: 1    4    9   16   25   36   49   64   81  100  121  144  169
The A(12,3) = 16 orderless factorizations of 12 with 2 levels of parentheses:
  ((2*2*3))          ((2*6))      ((3*4))      ((12))
  ((2)*(2*3))        ((2)*(6))    ((3)*(4))
  ((3)*(2*2))        ((2))*((6))  ((3))*((4))
  ((2))*((2*3))
  ((2)*(2)*(3))
  ((3))*((2*2))
  ((2))*((2)*(3))
  ((3))*((2)*(2))
  ((2))*((2))*((3))

Columns: A001055 (k=1), A050336 (k=2), A050338 (k=3), A050340 (k=4).
Rows: A000027, A000217, A000290, etc.
Cf. A096751, A141268, A144150, A213427, A255906, A281113, A290353, A292504, A317145, A318564, A318565, A318812, A323718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
lev[n_,k_]:=If[k==0,{n},Join@@Table[Union[Sort/@Tuples[lev[#,k-1]&/@fac]],{fac,facs[n]}]];
Table[Length[lev[sum-k,k]],{sum,12},{k,0,sum-1}]

A007715 Number of 5-leaf rooted trees with n levels.

Original entry on oeis.org

1, 7, 27, 75, 170, 336, 602, 1002, 1575, 2365, 3421, 4797, 6552, 8750, 11460, 14756, 18717, 23427, 28975, 35455, 42966, 51612, 61502, 72750, 85475, 99801, 115857, 133777, 153700, 175770, 200136, 226952, 256377, 288575, 323715, 361971, 403522, 448552, 497250

Offset: 1

N. J. A. Sloane, Simon Plouffe

			a(7) = 7*28 - (7*0+4*1+1*3-2*6-5*10-8*15-11*21) = 602. - _Bruno Berselli_, Jun 22 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
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.
Index entries for sequences related to rooted trees
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Row n=5 of A290353.

[n*(n+1)*(5*n^2+n+6)/24: n in [1..45]]; // Vincenzo Librandi, Jul 21 2011

Table[n(n+1)(5n^2+n+6)/24,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,7,27,75,170},40] (* Harvey P. Dale, Jul 20 2011 *)

Expansion of x*(1+2x+2x^2)/(1-x)^5.
a(n) = n*(n+1)*(5*n^2+n+6)/24. - T. D. Noe, Feb 09 2007
a(1)=1, a(2)=7, a(3)=27, a(4)=75, a(5)=170, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Jul 20 2011
a(n) = n*A000217(n) - sum((n-3*i)*A000217(i), i=0..n-1). - Bruno Berselli, Jun 22 2013

A290359 The tenth Euler transform of the sequence with g.f. 1+x.

Original entry on oeis.org

1, 1, 10, 55, 385, 2365, 15367, 96613, 611644, 3832477, 23970089, 149170604, 925530638, 5722654098, 35282873191, 216928671076, 1330360845060, 8139139896353, 49683631194244, 302640125458942, 1839793530751731, 11163107720200726, 67610680329079976

Offset: 0

Alois P. Heinz, Jul 28 2017

nonn

Also the number of 10-level rooted trees with n leaves. All n leaves are in level 10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees

Column k=10 of A290353.

Cf. A290358.

with(numtheory):
b:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(b(d, k-1)*d, d=divisors(j))*b(n-j, k), j=1..n)/n))
    end:
a:= n-> b(n, 10):
seq(a(n), n=0..30);

b[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[b[d, k - 1]*d, {d, Divisors[j]}] b[n - j, k], {j, n}]/n]]; Table[b[n, 10], {n, 0, 30}] (* Indranil Ghosh, Jul 30 2017 *)

G.f.: Product_{j>0} 1/(1-x^j)^A290358(j).

A290360 Number of 6-leaf rooted trees with n levels.

Original entry on oeis.org

0, 1, 11, 58, 206, 571, 1337, 2772, 5244, 9237, 15367, 24398, 37258, 55055, 79093, 110888, 152184, 204969, 271491, 354274, 456134, 580195, 729905, 909052, 1121780, 1372605, 1666431, 2008566, 2404738, 2861111, 3384301, 3981392, 4659952, 5428049, 6294267

Offset: 0

Alois P. Heinz, Jul 28 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row n=6 of A290353.

a:= n-> ((((4*n+5)*n+10)*n+10)*n+1)*n/30:
seq(a(n), n=0..40);

LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,11,58,206,571},40] (* Harvey P. Dale, Aug 22 2019 *)

G.f.: (3*x+1)*(x+1)^2*x / (x-1)^6.

a(n) = (4*n^5+5*n^4+10*n^3+10*n^2+n)/30.

A290361 Number of 7-leaf rooted trees with n levels.

Original entry on oeis.org

0, 1, 15, 111, 518, 1789, 5026, 12166, 26328, 52221, 96613, 168861, 281502, 450905, 697984, 1048972, 1536256, 2199273, 3085467, 4251307, 5763366, 7699461, 10149854, 13218514, 17024440, 21703045, 27407601, 34310745, 42606046, 52509633, 64261884, 78129176

Offset: 0

Alois P. Heinz, Jul 28 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Row n=7 of A290353.

a:= n-> (((((61*n+69)*n+145)*n+195)*n+154)*n+96)*n/6!:
seq(a(n), n=0..40);

G.f.: -(4*x^4+21*x^3+27*x^2+8*x+1)*x / (x-1)^7.

a(n) = (61*n^6+69*n^5+145*n^4+195*n^3+154*n^2+96*n)/6!.

A290362 Number of 8-leaf rooted trees with n levels.

Original entry on oeis.org

0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, 297633, 611644, 1175845, 2138500, 3711279, 6187767, 9965276, 15570232, 23687409, 35193282, 51193771, 73066648, 102508879, 141589173, 192806010, 259151420, 344180785, 452088936, 587792817, 757020988, 966410239

Offset: 0

Alois P. Heinz, Jul 28 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Row n=8 of A290353.

a:= n-> ((((((272*n+273)*n+749)*n+1365)*n+1043)*n+882)*n+456)*n/7!:
seq(a(n), n=0..40);

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{0,1,22,223,1344,5727,19193,54046},30] (* Harvey P. Dale, Oct 16 2017 *)

G.f.: (5*x^5+57*x^4+120*x^3+75*x^2+14*x+1)*x / (x-1)^8.

a(n) = (272*n^7+273*n^6+749*n^5+1365*n^4+1043*n^3+882*n^2+456*n)/7!.

A290363 Number of 9-leaf rooted trees with n levels.

Original entry on oeis.org

0, 1, 30, 424, 3357, 17836, 71769, 236093, 667335, 1676364, 3832477, 8113347, 16112746, 30319341, 54481246, 94072398, 156878210, 253719339, 399333792, 613438978, 921996699, 1358705458, 1966745847, 2800806163, 3929416785, 5437623230, 7430029191, 10034242245

Offset: 0

Alois P. Heinz, Jul 28 2017

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
Index entries for sequences related to rooted trees
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row n=9 of A290353.

a:= n-> (((((((1385*n+1124)*n+4018)*n+6776)*n+7945)*n
        +9716)*n+6812)*n+2544)*n/8!:
seq(a(n), n=0..40);

CoefficientList[Series[-(8*x^6 + 135*x^5 + 493*x^4 + 537*x^3 + 190*x^2 + 21*x + 1)*x/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Oct 14 2023 *)

G.f.: -(8*x^6+135*x^5+493*x^4+537*x^3+190*x^2+21*x+1)*x / (x-1)^9.

a(n) = (1385*n^8 +1124*n^7 +4018*n^6 +6776*n^5 +7945*n^4 +9716*n^3 +6812*n^2 +2544*n)/8!.

cp's OEIS Frontend

A290356 The seventh Euler transform of the sequence with g.f. 1+x.

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A290357 The eighth Euler transform of the sequence with g.f. 1+x.

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A290358 The ninth Euler transform of the sequence with g.f. 1+x.

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Maple

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A323719 Array read by antidiagonals upwards where A(n, k) is the number of orderless factorizations of n with k - 1 levels of parentheses.

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A007715 Number of 5-leaf rooted trees with n levels.

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A290359 The tenth Euler transform of the sequence with g.f. 1+x.

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Maple

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A290360 Number of 6-leaf rooted trees with n levels.

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A290361 Number of 7-leaf rooted trees with n levels.

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