A255704 -id:A255704 - cp's OEIS frontend

A255636 Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 3, 4, 0, 1, 1, 2, 4, 7, 8, 0, 1, 1, 2, 4, 8, 15, 17, 0, 1, 1, 2, 4, 9, 18, 35, 36, 0, 1, 1, 2, 4, 9, 19, 43, 81, 79, 0, 1, 1, 2, 4, 9, 20, 46, 102, 195, 175, 0, 1, 1, 2, 4, 9, 20, 47, 110, 251, 473, 395, 0

Offset: 1

Alois P. Heinz, Feb 28 2015

Any rootward k-node path starting at a leaf contains the root or a branching node.

			:    o      o        o      o    o       o      o    o
:  /(|)\    |       / \    /|\   |       |     / \   |
: o ooo o   o      o   o  o o o  o       o    o   o  o
:         /( )\   /|\    / \     |      / \   |      |
:        o o o o o o o  o   o    o     o   o  o      o
:                               /|\   / \    / \     |
:                              o o o o   o  o   o    o
: A(6,2) = 8                                        / \
:                                                  o   o
Square array A(n,k) begins:
  1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   2,   2,   2,   2,   2,   2,   2, ...
  0,   2,   3,   4,   4,   4,   4,   4,   4,   4, ...
  0,   4,   7,   8,   9,   9,   9,   9,   9,   9, ...
  0,   8,  15,  18,  19,  20,  20,  20,  20,  20, ...
  0,  17,  35,  43,  46,  47,  48,  48,  48,  48, ...
  0,  36,  81, 102, 110, 113, 114, 115, 115, 115, ...
  0,  79, 195, 251, 273, 281, 284, 285, 286, 286, ...
  0, 175, 473, 625, 684, 706, 714, 717, 718, 719, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Columns k=1-10 give: A063524, A002955, A052321, A052327, A052328, A052329, A255637, A255638, A255639, A255640.
Main diagonal gives A000081.
Cf. A255704.

with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
A:= (n, k)-> g(n-1, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);

g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[d*(g[d - 1, k] - If[d == k, 1, 0]), {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n]; A[n_, k_] := g[n - 1, k]; Table[Table[A[n, 1 + d - n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A255705 Number of 2n+1-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals n+1.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 167, 465, 1306, 3681, 10422, 29597, 84313, 240757, 689035, 1975753, 5675145, 16326198, 47032200, 135658367, 391733593, 1132357784, 3276330780, 9487885056, 27497891241, 79753806451, 231474005120, 672250119756, 1953523496677, 5680002466125

Offset: 0

Alois P. Heinz, Mar 02 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A255636, A255704, A318754, A318758.

with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
a:= a-> g(2*n, n+1) -`if`(n=0, 0, g(2*n, n)):
seq(a(n), n=0..40);

g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[# - 1, k] - If[# == k, 1, 0]) &]*g[n - j, k], {j, 1, n}]/n];
a[n_] :=  g[2n, n+1] - If[n == 0, 0, g[2n, n]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

a(n) = A255704(2*n+1,n+1).
a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.955765285651994974714817524... and c = 0.70755335886284109851526791506579... . - Vaclav Kotesovec, Feb 28 2016
a(n) = A318754(2n+2,n+1) = A318758(2n+2,n+1). - Alois P. Heinz, Sep 02 2018

A318899 Number of n-node rooted trees in which three equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 7, 18, 45, 116, 298, 776, 2025, 5322, 14030, 37155, 98685, 262961, 702497, 1881475, 5050140, 13583622, 36605565, 98821445, 267220361, 723704046, 1962830775, 5330900916, 14497096134, 39472561082, 107601053713, 293643574776, 802203904616, 2193758306687

Offset: 3

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..2209

Column k=3 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(3):
seq(a(n), n=3..35);

A318900 Number of n-node rooted trees in which four equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 21, 56, 152, 413, 1131, 3113, 8603, 23861, 66386, 185190, 517807, 1450836, 4072474, 11450141, 32240466, 90901507, 256605905, 725176409, 2051455548, 5808817638, 16462274560, 46691893949, 132531324274, 376443234567, 1069953746735, 3042969864371

Offset: 4

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..2158

Column k=4 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(4):
seq(a(n), n=4..35);

A318901 Number of n-node rooted trees in which five equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 163, 450, 1254, 3505, 9846, 27740, 78412, 222175, 630993, 1795524, 5118371, 14613211, 41780583, 119605948, 342793785, 983487162, 2824375723, 8118198640, 23353430828, 67230869761, 193682427149, 558333769713, 1610500116245, 4648080322122

Offset: 5

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..2143

Column k=5 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(5):
seq(a(n), n=5..35);

A318902 Number of n-node rooted trees in which six equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 166, 461, 1291, 3629, 10246, 29020, 82448, 234818, 670288, 1917054, 5492422, 15760308, 45286760, 130293687, 375293797, 1082109082, 3123088057, 9021467714, 26080762653, 75454838291, 218451043430, 632849337956, 1834453919208, 5320549626803

Offset: 6

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..2138

Column k=6 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(6):
seq(a(n), n=6..36);

A318903 Number of n-node rooted trees in which seven equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 167, 464, 1302, 3666, 10370, 29421, 83736, 238891, 683088, 1956968, 5616281, 16142818, 46463814, 133903792, 386336345, 1115804329, 3225691950, 9333321576, 27027053245, 78322024353, 227126864470, 659069928758, 1913612752613, 5559288014180

Offset: 7

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..2137

Column k=7 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(7):
seq(a(n), n=7..37);

A318904 Number of n-node rooted trees in which eight equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 167, 465, 1305, 3677, 10407, 29545, 84137, 240180, 687169, 1969805, 5656352, 16267296, 46848655, 135089324, 389976522, 1126951399, 3259744738, 9437132452, 27342937037, 79281644947, 230037790501, 667888276997, 1940294941620, 5639933760363

Offset: 8

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..2137

Column k=8 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(8):
seq(a(n), n=8..38);

A318905 Number of n-node rooted trees in which nine equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 167, 465, 1306, 3680, 10418, 29582, 84261, 240581, 688458, 1973887, 5669197, 16307404, 46973290, 135474784, 391164385, 1130600056, 3270921899, 9471289877, 27447106048, 79598738454, 231001452777, 670812581033, 1949157218646, 5666759197092

Offset: 9

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..2137

Column k=9 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(9):
seq(a(n), n=9..39);

A318906 Number of n-node rooted trees in which ten equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 167, 465, 1306, 3681, 10421, 29593, 84298, 240705, 688859, 1975176, 5673279, 16320250, 47013406, 135599456, 391550002, 1131788538, 3274572887, 9482475518, 27481293566, 79703012121, 231318904534, 671777453618, 1952085567096, 5675634863875

Offset: 10

Alois P. Heinz, Sep 05 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..2138

Column k=10 of A255704.

g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
a:= n-> (k-> g(n-1, k) -g(n-1, k-1))(10):
seq(a(n), n=10..40);

cp's OEIS Frontend

A255636 Number A(n,k) of n-node rooted trees with a forbidden limb of length k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A255705 Number of 2n+1-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals n+1.

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A318899 Number of n-node rooted trees in which three equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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Maple

A318900 Number of n-node rooted trees in which four equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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A318901 Number of n-node rooted trees in which five equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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A318902 Number of n-node rooted trees in which six equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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Maple

A318903 Number of n-node rooted trees in which seven equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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A318904 Number of n-node rooted trees in which eight equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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Maple

A318905 Number of n-node rooted trees in which nine equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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Maple

A318906 Number of n-node rooted trees in which ten equals the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root.

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