cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A291529 Number F(n,h,t) of forests of t (unlabeled) rooted identity trees with n vertices such that h is the maximum of 0 and the tree heights; triangle of triangles F(n,h,t), n>=0, h=0..n, t=0..n-h, read by layers, then by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 5, 1, 0, 0, 5, 4, 0, 0, 4, 1, 0, 1, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 25 2017

Keywords

Comments

Positive row sums per layer (and - with a different offset - positive elements in column t=1) give A227819.
Positive column sums per layer give A227774.

Examples

			n h\t: 0 1 2 3 4 5 : A227819 : A227774   : A004111
-----+-------------+---------+-----------+--------
0 0  : 1           :         :           : 1
-----+-------------+---------+-----------+--------
1 0  : 0 1         :       1 : .         :
1 1  : 0           :         : 1         : 1
-----+-------------+---------+-----------+--------
2 0  : 0 0 0       :       0 : . .       :
2 1  : 0 1         :       1 : .         :
2 2  : 0           :         : 1 0       : 1
-----+-------------+---------+-----------+--------
3 0  : 0 0 0 0     :       0 : . . .     :
3 1  : 0 0 1       :       1 : . .       :
3 2  : 0 1         :       1 : .         :
3 3  : 0           :         : 1 1 0     : 2
-----+-------------+---------+-----------+--------
4 0  : 0 0 0 0 0   :       0 : . . . .   :
4 1  : 0 0 0 0     :       0 : . . .     :
4 2  : 0 1 1       :       2 : . .       :
4 3  : 0 1         :       1 : .         :
4 4  : 0           :         : 2 1 0 0   : 3
-----+-------------+---------+-----------+--------
5 0  : 0 0 0 0 0 0 :       0 : . . . . . :
5 1  : 0 0 0 0 0   :       0 : . . . .   :
5 2  : 0 0 2 0     :       2 : . . .     :
5 3  : 0 2 1       :       3 : . .       :
5 4  : 0 1         :       1 : .         :
5 5  : 0           :         : 3 3 0 0 0 : 6
-----+-------------+---------+-----------+--------
		

Crossrefs

Programs

  • Maple
    b:= proc(n, i, t, h) option remember; expand(`if`(n=0 or h=0 or i=1,
          `if`(n<2, x^(t*n), 0), b(n, i-1, t, h)+add(x^(t*j)*binomial(
           b(i-1$2, 0, h-1), j)*b(n-i*j, i-1, t, h), j=1..n/i)))
        end:
    g:= (n, h)-> b(n$2, 1, h)-`if`(h=0, 0, b(n$2, 1, h-1)):
    F:= (n, h, t)-> coeff(g(n, h), x, t):
    seq(seq(seq(F(n, h, t), t=0..n-h), h=0..n), n=0..10);
  • Mathematica
    b[n_, i_, t_, h_] := b[n, i, t, h] = Expand[If[n == 0 || h == 0 || i == 1, If[n < 2, x^(t*n), 0], b[n, i - 1, t, h] + Sum[x^(t*j)*Binomial[b[i - 1, i - 1, 0, h - 1], j]*b[n - i*j, i - 1, t, h], {j, 1, n/i}]]];
    g[n_, h_] := b[n, n, 1, h] - If[h == 0, 0, b[n, n, 1, h - 1]];
    F[n_, h_, t_] := Coefficient[g[n, h], x, t];
    Table[F[n, h, t], {n, 0, 10}, {h, 0, n}, {t, 0, n - h}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *)

Formula

Sum_{d=0..n} Sum_{i=0..d} F(n,i,d-i) = A004111(n+1).
Sum_{h=0..n} Sum_{t=0..n-h} t * F(n,h,t) = A291532(n).
Sum_{h=0..n-2} Sum_{t=1..n-1-h} (h+1) * F(n-1,h,t) = A291559(n).
F(n,0,0) = A000007(n).

A227806 Number of rooted identity trees with n nodes and exactly 2 subtrees from the root.

Original entry on oeis.org

1, 1, 3, 5, 11, 22, 49, 104, 232, 513, 1159, 2619, 5989, 13734, 31729, 73555, 171377, 400631, 940104, 2212542, 5222932, 12360976, 29327260, 69735757, 166170966, 396727768, 948897250, 2273409345, 5455374972, 13110384631, 31550978034, 76029236983, 183437066950
Offset: 4

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Examples

			:    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      :
:         :         :                    |      :
:  n=4    :  n=5    :  n=6               o      :
		

Crossrefs

Column k=2 of A227774.

Formula

a(n) ~ c * d^n / n^(3/2), where d = A246169 = 2.51754035263200389079535459846344... and c = 0.14400421547102520752812171064737721... - Vaclav Kotesovec, Jun 07 2021

A291532 Number of trees in all forests of (unlabeled) rooted identity trees with n vertices.

Original entry on oeis.org

0, 1, 1, 3, 4, 9, 19, 40, 84, 186, 413, 922, 2082, 4733, 10831, 24928, 57648, 133923, 312393, 731328, 1717784, 4047111, 9561517, 22647521, 53770164, 127941813, 305046676, 728688803, 1743752229, 4179697971, 10034077377, 24123567285, 58076419495, 139996849639
Offset: 0

Views

Author

Alois P. Heinz, Aug 25 2017

Keywords

Examples

			a(4) = 4:
:   o   :   o  o   :     o     :
:   |   :   |      :    / \    :
:   o   :   o      :   o   o   :
:   |   :   |      :   |       :
:   o   :   o      :   o       :
:   |   :          :           :
:   o   :          :           :
:       :          :           :
		

Crossrefs

Programs

  • Maple
    b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
           `if`(n<2, x^(t*n), 0), b(n, i-1, t)+add(binomial(
           b(i-1$2, 0), j)*x^(t*j)*b(n-i*j, i-1, t), j=1..n/i)))
        end:
    a:= n-> (p-> add(i*coeff(p,x,i), i=1..degree(p)))(b(n$2, 1)):
    seq(a(n), n=0..35);
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0 || i == 1,
         If[n < 2, x^(t*n), 0], b[n, i - 1, t] + Sum[Binomial[
         b[i - 1, i - 1, 0], j]*x^(t*j)*b[n - i*j, i - 1, t], {j, 1, n/i}]]];
    a[n_] := Function[p, Sum[i*Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][
         b[n, n, 1]];
    Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=1} k * A227774(n+1,k).
a(n) = Sum_{h=0..n} Sum_{t=0..n-h} t * A291529(n,h,t).

A227807 Number of rooted identity trees with n nodes and exactly 3 subtrees from the root.

Original entry on oeis.org

1, 2, 5, 12, 28, 65, 152, 351, 818, 1907, 4460, 10453, 24581, 57922, 136860, 324088, 769188, 1829282, 4359048, 10406106, 24885240, 59607192, 142996100, 343540150, 826468374, 1990841497, 4801534221, 11593858684, 28025623617, 67816875039, 164268161404
Offset: 7

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Examples

			:    o    :      o      o    :
:   /|\   :     /|\    /|\   :
:  o o o  :    o o o  o o o  :
:  | |    :   /| |    | |    :
:  o o    :  o o o    o o    :
:  |      :  |        |      :
:  o      :  o        o      :
:         :           |      :
:  n=7    :  n=8      o      :
		

Crossrefs

Column k=3 of A227774.

A227808 Number of rooted identity trees with n nodes and exactly 4 subtrees from the root.

Original entry on oeis.org

2, 4, 13, 34, 91, 225, 571, 1403, 3466, 8468, 20703, 50389, 122673, 298090, 724433, 1759595, 4274819, 10385533, 25240007, 61358410, 149226407, 363077813, 883818374, 2152462147, 5244790003, 12786162151, 31187100089, 76108058977, 185825963102, 453940832229
Offset: 11

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Examples

			:        o            o      :
:      /( )\        /( )\    :
:     / | | \      / | | \   :
:    o  o o  o    o  o o  o  :
:   /|  | |       |  | |     :
:  o o  o o       o  o o     :
:  |    |         |  |       :
:  o    o         o  o       :
:                 |          :
:  n=11           o          :
		

Crossrefs

Column k=4 of A227774.

A227809 Number of rooted identity trees with n nodes and exactly 5 subtrees from the root.

Original entry on oeis.org

1, 6, 18, 57, 165, 460, 1242, 3295, 8589, 22169, 56675, 143941, 363527, 914253, 2291088, 5725579, 14275461, 35526939, 88279211, 219088566, 543169235, 1345516437, 3330785029, 8240718421, 20379364252, 50380569630, 124513186628, 307663311638, 760095010218
Offset: 15

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Examples

			: n=15     o        :
:       / /|\ \     :
:      / ( | ) \    :
:     o  o o o  o   :
:    /|  | | |      :
:   o o  o o o      :
:   |    | |        :
:   o    o o        :
:        |          :
:        o          :
		

Crossrefs

Column k=5 of A227774.

A227810 Number of rooted identity trees with n nodes and exactly 6 subtrees from the root.

Original entry on oeis.org

3, 12, 52, 168, 537, 1591, 4605, 12878, 35434, 95686, 255451, 674351, 1765821, 4589927, 11862112, 30500143, 78095203, 199230786, 506693631, 1285172861, 3252142655, 8212876975, 20703986186, 52112143876, 130988626702, 328857650246, 824753010860, 2066496137418
Offset: 20

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Crossrefs

Column k=6 of A227774.

A227811 Number of rooted identity trees with n nodes and exactly 7 subtrees from the root.

Original entry on oeis.org

3, 20, 88, 334, 1163, 3756, 11615, 34655, 100653, 286033, 798924, 2199461, 5983705, 16115809, 43036487, 114089624, 300554336, 787451267, 2053299653, 5331645252, 13793247146, 35567344417, 91447996464, 234515163492, 600012257027, 1531951193292, 3904053947702
Offset: 25

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Crossrefs

Column k=7 of A227774.

A227812 Number of rooted identity trees with n nodes and exactly 8 subtrees from the root.

Original entry on oeis.org

1, 18, 93, 431, 1664, 5982, 20059, 64473, 199450, 600429, 1765198, 5093281, 14461282, 40514663, 112199665, 307665178, 836401955, 2256766751, 6048944297, 16118777953, 42729253427, 112746836489, 296263589093, 775585850534, 2023567436543, 5263578982246
Offset: 30

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Crossrefs

Column k=8 of A227774.

A227813 Number of rooted identity trees with n nodes and exactly 9 subtrees from the root.

Original entry on oeis.org

6, 57, 331, 1534, 6271, 23458, 82376, 275587, 887706, 2773137, 8448475, 25203966, 73869495, 213244665, 607595613, 1711636481, 4774000801, 13198942637, 36208850834, 98645557887, 267081435051, 719094529584, 1926377343196, 5137083969024, 13642483285790
Offset: 36

Views

Author

Alois P. Heinz, Jul 31 2013

Keywords

Crossrefs

Column k=9 of A227774.
Showing 1-10 of 11 results. Next