A316074 -id:A316074 - cp's OEIS frontend

A004111 Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).

0, 1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, 33209, 76851, 178618, 416848, 976296, 2294224, 5407384, 12780394, 30283120, 71924647, 171196956, 408310668, 975662480, 2335443077, 5599508648, 13446130438, 32334837886, 77863375126, 187737500013, 453203435319, 1095295264857, 2649957419351

Offset: 0

N. J. A. Sloane

The nodes are unlabeled.
There is a natural correspondence between rooted identity trees and finitary sets (sets whose transitive closure is finite); each node represents a set, with the children of that node representing the members of that set. When the set corresponding to an identity tree is written out using braces, there is one set of braces for each node of the tree; thus a(n) is also the number of sets that can be made using n pairs of braces. - Franklin T. Adams-Watters, Oct 25 2011
Shifts left under WEIGH transform. - Franklin T. Adams-Watters, Jan 17 2007
Is this the sequence mentioned in the middle of page 355 of Motzkin (1948)? - N. J. A. Sloane, Jul 04 2015. Answer from David Broadhurst, Apr 06 2022: The answer is No. Motzkin was considering a sequence asymptotic to Catalan(n)/(4*n), namely A006082, which begins 1, 1, 1, 2, 3, 6, 12, 27, ... but he miscalculated and got 1, 1, 1, 2, 3, 6, 12, 25, ... instead! - N. J. A. Sloane, Apr 06 2022

			The 2 identity trees with 4 nodes are:
     O    O
    / \   |
   O   O  O
       |  |
       O  O
          |
          O
These correspond to the sets {{},{{}}} and {{{{}}}}.
G.f.: x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 12*x^7 + 25*x^8 + 52*x^9 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 330.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 64, Eq. (3.3.15); p. 80, Problem 3.10.
D. E. Knuth, Fundamental Algorithms, 3rd Ed., 1997, pp. 386-388.
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 = 0..2500 (first 201 terms from T. D. Noe)
Joerg Arndt, All identity trees for n = 1..11.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
A. Genitrini, Full asymptotic expansion for Polya structures, arXiv:1605.00837 [math.CO], May 03 2016, p. 8.
Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017-2018; Discrete Math., 341 (2018), 896-911.
Frank Harary and Geert Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162.
F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503.
F. Harary, R. W. Robinson and A. J. Schwenk, Corrigenda: Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A 41 (1986), p. 325.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 56.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
N. J. A. Sloane, Sketch showing trees with 2 through 6 nodes.
Index entries for sequences related to rooted trees

Cf. A000009, A000081, A000220, A196118, A196154, A196161, A227819.
Cf. A027750, A035056, A246169.
Column k=1 of A255517, A316074, A316101, A318757.

import Data.List (genericIndex)
a004111 = genericIndex a004111_list
a004111_list = 0 : 1 : f 1 [1] where
   f x zs = y : f (x + 1) (y : zs) where
            y = (sum $ zipWith (*) zs $ map g [1..]) `div` x
   g k = sum $ zipWith (*) (map (((-1) ^) . (+ 1)) $ reverse divs)
                           (zipWith (*) divs $ map a004111 divs)
                           where divs = a027750_row k
-- Reinhard Zumkeller, Apr 29 2014

A004111 := proc(n)
        spec := [ A, {A=Prod(Z,PowerSet(A))} ]:
        combstruct[count](spec, size=n) ;
end proc:
# second Maple program:
with(numtheory):
a:= proc(n) a(n):= `if`(n<2, n, add(a(n-k)*add(a(d)*d*
       (-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 15 2014

s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (* Robert A. Russell *)
a[ n_] := If[ n < 2, Boole[n == 1], Nest[ CoefficientList[ Normal[ Times @@ (Table[1 + x^k, {k, Length@#}]^#) + x O[x]^Length@#], x] &, {}, n - 1][[n]]]; (* Michael Somos, Jul 10 2014 *)
a[n_] := a[n] = Sum[a[n-k]*Sum[a[d]*d*(-1)^(k/d+1),{d, Divisors[k]}], {k, 1, n-1}]/(n-1); a[0]=0; a[1]=1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 02 2015 *)

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d * A[d]) * A[n-k+1] ) );
concat([0], A)
\\ Joerg Arndt, Jul 10 2014

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) d*a(d) ) * a(n-k+1). - Mitchell Harris, Dec 02 2004
G.f. satisfies A(x) = x*exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...). [Harary and Prins]
Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = A246169 = 2.51754035263200389079535..., c = 0.3625364233974198712298411097408713812865256408189512533230825639621448038... . - Vaclav Kotesovec, Aug 22 2014, updated Dec 26 2020

A007560 Number of planted identity trees where non-root, non-leaf nodes an even distance from root are of degree 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 10, 17, 29, 51, 89, 159, 284, 512, 930, 1692, 3101, 5698, 10515, 19464, 36143, 67296, 125622, 235050, 440756, 828142, 1558955, 2939761, 5552744, 10504222, 19899760, 37750091, 71704061, 136361602, 259618770, 494821629, 944074665

Offset: 1

N. J. A. Sloane

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
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A007562.

Column k=2 of A316074.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-2, n-2)):
seq(a(n), n=1..50);  # Alois P. Heinz, May 19 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<2, n, b[n-2, n-2]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)

Shifts 2 places left under weigh transform.
a(n) ~ c * d^n / n^(3/2), d = 1.983229991815043367273184141..., c = 0.5857451140002020594085469... . - Vaclav Kotesovec, Aug 25 2014
G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

Better description from Christian G. Bower, May 15 1998

A316101 Sequence a_k of column k shifts left when Weigh transform is applied k times with a_k(n) = n for n<2; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 3, 3, 1, 0, 1, 1, 1, 4, 6, 6, 1, 0, 1, 1, 1, 5, 10, 16, 12, 1, 0, 1, 1, 1, 6, 15, 32, 43, 25, 1, 0, 1, 1, 1, 7, 21, 55, 105, 120, 52, 1, 0, 1, 1, 1, 8, 28, 86, 210, 356, 339, 113, 1, 0, 1, 1, 1, 9, 36, 126, 371, 826, 1227, 985, 247, 1

Offset: 0

Alois P. Heinz, Jun 24 2018

			Square array A(n,k) begins:
  0,  0,   0,   0,   0,    0,    0,    0,    0, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  2,   3,   4,   5,    6,    7,    8,    9, ...
  1,  3,   6,  10,  15,   21,   28,   36,   45, ...
  1,  6,  16,  32,  55,   86,  126,  176,  237, ...
  1, 12,  43, 105, 210,  371,  602,  918, 1335, ...
  1, 25, 120, 356, 826, 1647, 2961, 4936, 7767, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Columns k=0-10 give: A057427, A004111, A007561, A316103, A316104, A316105, A316106, A316107, A316108, A316109, A316110.
Rows include (offsets may differ): A000004, A000012, A000027, A000217, A134465.
Main diagonal gives A316102.
Cf. A144042, A316074.

wtr:= proc(p) local b; b:= proc(n, i) option remember;
       `if`(n=0, 1, `if`(i<1, 0, add(binomial(p(i), j)*
         b(n-i*j, i-1), j=0..n/i))) end: j-> b(j$2)
      end:
g:= proc(k) option remember; local b, t; b[0]:= j->
     `if`(j<2, j, b[k](j-1)); for t to k do
       b[t]:= wtr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, d-n), n=0..d), d=0..14);

wtr[p_] := Module[{b}, b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[p[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]]; b[#, #]&];
g[k_] := g[k] = Module[{b, t}, b[0][j_] := If[j < 2, j, b[k][j - 1]]; For[ t = 1, t <= k + 1, t++, b[t] = wtr[b[t - 1]]]; b[0]];
A[n_, k_] := g[k][n];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 3, 2, 1, 1, 20, 6, 3, 2, 1, 1, 48, 10, 5, 3, 2, 1, 1, 115, 20, 8, 5, 3, 2, 1, 1, 286, 36, 14, 7, 5, 3, 2, 1, 1, 719, 72, 23, 12, 7, 5, 3, 2, 1, 1, 1842, 137, 40, 18, 11, 7, 5, 3, 2, 1, 1, 4766, 275, 69, 30, 16, 11, 7, 5, 3, 2, 1, 1, 12486, 541, 121, 47, 25, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Sep 07 2008

			T(5,1) = ([1,2,4]*[1,1,4] + [1]*[1]*4 + [1,2]*[1,1]*2 + [1,3]*[1,2]*1)/4 = 36/4 = 9.
Triangle begins:
    1;
    1,  1;
    2,  1,  1;
    4,  2,  1,  1;
    9,  3,  2,  1, 1;
   20,  6,  3,  2, 1, 1;
   48, 10,  5,  3, 2, 1, 1;
  115, 20,  8,  5, 3, 2, 1, 1;
  286, 36, 14,  7, 5, 3, 2, 1, 1;
  719, 72, 23, 12, 7, 5, 3, 2, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.
T(2n,n) gives A000041(n).
Cf. A316074.

etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,
         add(add(d*p(d, k), d=numtheory[divisors](j))*
         procname(n-j, k), j=1..n)/n)
       end end:
B:= etrk(T):
T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):
seq(seq(T(n, k), k=1..n), n=1..14);

etrk[p_] := Module[{f}, f[n_, k_] := f[n, k] = If[n == 0, 1, (Sum[Sum[d*p[d, k], {d, Divisors[j]}]*f[n-j, k], {j, 1, n-1}] + Sum[d*p[d, k], {d, Divisors[n]}])/n]; f]; b = etrk[t]; t[n_, k_] := If[n <= k, If[n == 0, 0, 1], b[n-k, k]]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

A316075 Sequence shifts left three places under Weigh transform with a(n) = signum(n) for n<3.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 5, 7, 10, 17, 26, 40, 65, 104, 166, 272, 442, 720, 1186, 1954, 3222, 5346, 8881, 14778, 24668, 41254, 69088, 115959, 194925, 328123, 553200, 933944, 1578614, 2671656, 4526654, 7677736, 13035809, 22154806, 37687152, 64165838

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Column k=3 of A316074.

Cf. A057427, A218020.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (k-> `if`(n

G.f.: x + x^2 + x^3 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

A316076 Sequence shifts left four places under Weigh transform with a(n) = signum(n) for n<4.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 18, 26, 39, 58, 88, 133, 202, 308, 472, 725, 1116, 1725, 2669, 4141, 6437, 10027, 15643, 24448, 38265, 59979, 94143, 147953, 232799, 366725, 578311, 912913, 1442503, 2281429, 3611406, 5721528, 9071789, 14394837

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Column k=4 of A316074.

Cf. A057427, A218021.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (k-> `if`(n

G.f.: x + x^2 + x^3 + x^4 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

A316077 Sequence shifts left five places under Weigh transform with a(n) = signum(n) for n<5.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 20, 28, 40, 58, 84, 121, 176, 257, 376, 552, 812, 1196, 1767, 2615, 3877, 5757, 8562, 12751, 19018, 28400, 42461, 63554, 95234, 142845, 214473, 322310, 484793, 729797, 1099509, 1657761, 2501299, 3776681

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Column k=5 of A316074.

Cf. A057427, A218022.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (k-> `if`(n

G.f.: x + x^2 + x^3 + x^4 + x^5 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

A316078 Sequence shifts left six places under Weigh transform with a(n) = signum(n) for n<6.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 12, 16, 23, 31, 43, 61, 85, 118, 168, 236, 333, 475, 674, 958, 1369, 1956, 2796, 4013, 5758, 8270, 11905, 17148, 24720, 35693, 51572, 74573, 107957, 156399, 226740, 329013, 477738, 694148, 1009326, 1468526

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Column k=6 of A316074.

Cf. A057427, A218023.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (k-> `if`(n

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

A316079 Sequence shifts left seven places under Weigh transform with a(n) = signum(n) for n<7.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 11, 14, 19, 26, 35, 48, 65, 89, 122, 167, 230, 317, 438, 606, 840, 1167, 1622, 2260, 3151, 4398, 6148, 8601, 12046, 16888, 23697, 33283, 46783, 65814, 92654, 130539, 184039, 259639, 366533, 517749, 731781

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2500

Column k=7 of A316074.

Cf. A057427, A218024.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (k-> `if`(n

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

A316080 Sequence shifts left eight places under Weigh transform with a(n) = signum(n) for n<8.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 17, 22, 30, 40, 53, 71, 96, 128, 173, 233, 314, 425, 575, 780, 1060, 1442, 1964, 2680, 3658, 5000, 6840, 9366, 12835, 17604, 24164, 33194, 45632, 62775, 86417, 119038, 164077, 226287, 312261, 431138

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2500

Column k=8 of A316074.

Cf. A057427, A218025.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (k-> `if`(n

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

cp's OEIS Frontend

A004111 Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).

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A007560 Number of planted identity trees where non-root, non-leaf nodes an even distance from root are of degree 2.

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A316101 Sequence a_k of column k shifts left when Weigh transform is applied k times with a_k(n) = n for n<2; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

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A316075 Sequence shifts left three places under Weigh transform with a(n) = signum(n) for n<3.

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A316076 Sequence shifts left four places under Weigh transform with a(n) = signum(n) for n<4.

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Maple

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A316077 Sequence shifts left five places under Weigh transform with a(n) = signum(n) for n<5.

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Maple

Formula

A316078 Sequence shifts left six places under Weigh transform with a(n) = signum(n) for n<6.

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