A244530 -id:A244530 - cp's OEIS frontend

A244454 Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 7, 1, 0, 1, 0, 17, 2, 0, 0, 1, 0, 42, 4, 1, 0, 0, 1, 0, 105, 7, 2, 0, 0, 0, 1, 0, 267, 15, 2, 1, 0, 0, 0, 1, 0, 684, 28, 4, 2, 0, 0, 0, 0, 1, 0, 1775, 56, 7, 2, 1, 0, 0, 0, 0, 1, 0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 28 2014

T(1,0) = 1 by convention.

Sum_{i=2..n-1} T(n,i) = A001678(n+1) for n>1.

			The A000081(5) = 9 rooted trees with 5 nodes sorted by minimal outdegree of inner nodes are:
: 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 o o o     : o   o   :         :
: |  / \  |     |                       :         :         :
: o o   o o     o                       :         :         :
: |                                     :         :         :
: o                                     :         :         :
:                                       :         :         :
: ------------------1------------------ : ---2--- : ---4--- :
Thus row 5 = [0, 7, 1, 0, 1].
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,   1;
  0,    3,   0,  1;
  0,    7,   1,  0, 1;
  0,   17,   2,  0, 0, 1;
  0,   42,   4,  1, 0, 0, 1;
  0,  105,   7,  2, 0, 0, 0, 1;
  0,  267,  15,  2, 1, 0, 0, 0, 1;
  0,  684,  28,  4, 2, 0, 0, 0, 0, 1;
  0, 1775,  56,  7, 2, 1, 0, 0, 0, 0, 1;
  0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1;

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

Columns k=0-10 give: A063524, A244455, A244456, A244457, A244458, A244459, A244460, A244461, A244462, A244463, A244464.
Row sums give A000081.
Cf. A001678, A244372, A244530 (ordered unlabeled rooted trees).

b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0, t-j), k), j=0..n/i)))
    end:
T:= (n, k)-> b(n-1$2, k$2) -`if`(n=1 and k=0, 0, b(n-1$2, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]* b[n-i*j, i-1, Max[0, t-j], k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)

A244531 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 2.

Original entry on oeis.org

1, 0, 2, 5, 11, 28, 78, 201, 532, 1441, 3895, 10569, 28926, 79493, 219226, 607189, 1687880, 4706737, 13165215, 36929595, 103860429, 292808814, 827392709, 2342964435, 6647953886, 18898472568, 53818654942, 153518738980, 438602656951, 1254943919799, 3595714927194

Offset: 3

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=2 of A244530.

Cf. A244456.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 2$2) -b(n-1, 3$2):
seq(a(n), n=3..50);

b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t > n, 0, Sum[b[j - 1, k, k]*b[n - j, Max[0, t - 1], k], {j, 1, n}]]]; T[n_, k_] := b[n - 1, k, k] - If[n == 1 && k == 0, 0, b[n - 1, k + 1, k + 1]]; a[n_] := b[n - 1, 2, 2] - b[n - 1, 3, 3]; Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Recurrence: (n-2)*n*(n+1)*(31556*n^6 - 602602*n^5 + 4562565*n^4 - 17272550*n^3 + 33523297*n^2 - 29665770*n + 7578864)*a(n) = -2*(n-4)*n*(15778*n^6 - 93541*n^5 - 718683*n^4 + 7746097*n^3 - 25426183*n^2 + 35870760*n - 18623988)*a(n-1) + 2*(189336*n^9 - 4357178*n^8 + 42198478*n^7 - 222932639*n^6 + 692179375*n^5 - 1246825745*n^4 + 1121148607*n^3 - 95771898*n^2 - 622360656*n + 342066240)*a(n-2) + 4*(15778*n^9 - 301301*n^8 + 2556736*n^7 - 13524389*n^6 + 51959635*n^5 - 145042550*n^4 + 255185823*n^3 - 199177680*n^2 - 62590212*n + 146335680)*a(n-3) - 2*(n-4)*(63112*n^8 - 1252538*n^7 + 9554713*n^6 - 31464554*n^5 + 11620330*n^4 + 221568106*n^3 - 627283143*n^2 + 624591414*n - 146644560)*a(n-4) - 4*(n-5)*(n-4)*(504896*n^7 - 9428629*n^6 + 69275668*n^5 - 250040744*n^4 + 437755491*n^3 - 253595994*n^2 - 179277570*n + 187109352)*a(n-5) - 69*(n-6)*(n-5)*(n-4)*(31556*n^6 - 413266*n^5 + 2022895*n^4 - 4417190*n^3 + 3528357*n^2 + 989760*n - 1844640)*a(n-6). - Vaclav Kotesovec, Jul 02 2014

a(n) ~ 3^(n+1/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 02 2014

A244532 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 3.

Original entry on oeis.org

1, 0, 0, 3, 7, 8, 21, 55, 121, 265, 611, 1379, 3193, 7436, 17085, 39339, 91846, 214549, 500132, 1169267, 2743302, 6445797, 15167805, 35749961, 84390645, 199523566, 472429633, 1120012481, 2658525869, 6318368820, 15034189965, 35811690663, 85393261630

Offset: 4

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Vaclav Kotesovec, Recurrence (of order 9)

Column k=3 of A244530.

Cf. A244457.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 3$2) -b(n-1, 4$2):
seq(a(n), n=4..40);

b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t > n, 0, Sum[b[j - 1, k, k]*b[n - j, Max[0, t - 1], k], {j, 1, n}]]]; T[n_, k_] := b[n - 1, k, k] - If[n == 1 && k == 0, 0, b[n - 1, k + 1, k + 1]]; a[n_] := b[n - 1, 3, 3] - b[n - 1, 4, 4]; Table[a[n], {n, 4, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

a(n) ~ sqrt(sqrt(2)/4 - sqrt(154+112*sqrt(2))/56) * ((sqrt(13+16*sqrt(2))-1)/2)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 02 2014

A244533 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 4.

Original entry on oeis.org

1, 0, 0, 0, 4, 9, 10, 11, 34, 91, 196, 330, 636, 1377, 2976, 6061, 12199, 25186, 52767, 109066, 224964, 467605, 979056, 2042847, 4244986, 8844130, 18527956, 38878929, 81460220, 170576593, 357894472, 752544917, 1583579674, 3332453026, 7016669752, 14790212086

Offset: 5

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000
Vaclav Kotesovec, Recurrence (of order 11)

Column k=4 of A244530.

Cf. A244458.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 4$2) -b(n-1, 5$2):
seq(a(n), n=5..45);

b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t > n, 0, Sum[b[j - 1, k, k]*b[n - j, Max[0, t - 1], k], {j, 1, n}]]]; T[n_, k_] := b[n - 1, k, k] - If[n == 1 && k == 0, 0, b[n - 1, k + 1, k + 1]]; a[n_] := b[n - 1, 4, 4] - b[n - 1, 5, 5]; Table[a[n], {n, 5, 45}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 2.18452974131524781307797151868229485574758... is the root of the equation -229 - 36*d + 2*d^2 - 32*d^3 + 19*d^4 + 4*d^5 = 0, and c = 0.181069926661856899940163775713243367029404419526724... . - Vaclav Kotesovec, Jul 02 2014

A244534 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 11, 12, 13, 14, 50, 136, 289, 477, 703, 1255, 2611, 5489, 10902, 19712, 35455, 66651, 130014, 254737, 488041, 920461, 1741642, 3338360, 6453073, 12425997, 23780944, 45451155, 87224392, 168253246, 324863578, 625728091, 1202953325, 2314485753

Offset: 6

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=5 of A244530.

Cf. A244459.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 5$2) -b(n-1, 6$2):
seq(a(n), n=6..50);

b[n_, t_, k_] := b[n, t, k] = If[n == 0,
     If[MemberQ[{0, k}, t], 1, 0], If[t > n, 0, Sum[b[j-1, k, k]*
     b[n-j, Max[0, t-1], k], {j, n}]]];
a[n_] := b[n-1, 5, 5] - b[n-1, 6, 6];
Table[a[n], {n, 6, 50}] (* Jean-François Alcover, Aug 28 2021, after Maple code *)

A244535 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 6, 13, 14, 15, 16, 17, 69, 190, 400, 651, 946, 1288, 2186, 4425, 9126, 17811, 31654, 51997, 85841, 149916, 276056, 518089, 960466, 1718395, 3006963, 5269873, 9392821, 17032418, 31098198, 56432687, 101350402, 180978701, 323731177, 582832779

Offset: 7

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=6 of A244530.

Cf. A244460.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 6$2) -b(n-1, 7$2):
seq(a(n), n=7..50);

A244536 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 7, 15, 16, 17, 18, 19, 20, 91, 253, 529, 852, 1225, 1651, 2133, 3493, 6931, 14095, 27156, 47648, 77297, 118031, 182462, 300441, 527398, 954712, 1722370, 3015910, 5074611, 8342271, 13730760, 23036563, 39558564, 68974240, 120541276

Offset: 8

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=7 of A244530.

Cf. A244461.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 7$2) -b(n-1, 8$2):
seq(a(n), n=8..50);

A244537 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 8, 17, 18, 19, 20, 21, 22, 23, 116, 325, 676, 1080, 1540, 2059, 2640, 3286, 5240, 10241, 20604, 39305, 68286, 109705, 165946, 239629, 351898, 552311, 931070, 1633871, 2879668, 4951860, 8208631, 13094200, 20436400, 31939817, 50935060

Offset: 9

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=8 of A244530.

Cf. A244462.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 8$2) -b(n-1, 9$2):
seq(a(n), n=9..55);

A244538 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 19, 20, 21, 22, 23, 24, 25, 26, 144, 406, 841, 1335, 1891, 2512, 3201, 3961, 4795, 7491, 14467, 28861, 54626, 94160, 150101, 225337, 323016, 446556, 629454, 949486, 1545877, 2639756, 4558225, 7716325, 12629776, 19928727, 30372551

Offset: 10

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=9 of A244530.

Cf. A244463.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 9$2) -b(n-1, 10$2):
seq(a(n), n=10..60);

A244539 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 10.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 21, 22, 23, 24, 25, 26, 27, 28, 29, 175, 496, 1024, 1617, 2278, 3010, 3816, 4699, 5662, 6708, 10310, 19721, 39074, 73487, 125862, 199365, 297436, 423799, 582472, 777777, 1060410, 1547051, 2443649, 4072732, 6905106, 11528110

Offset: 11

Joerg Arndt and Alois P. Heinz, Jun 29 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..1000

Column k=10 of A244530.

Cf. A244464.

b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
a:= n-> b(n-1, 10$2) -b(n-1, 11$2):
seq(a(n), n=11..60);

cp's OEIS Frontend

A244454 Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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A244531 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 2.

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A244532 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 3.

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A244533 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 4.

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A244534 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 5.

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A244535 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 6.

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A244536 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 7.

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A244537 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 8.

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A244538 Number of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 9.

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