A000684 -id:A000684 - cp's OEIS frontend

A001227 Number of odd divisors of n.

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8

Offset: 1

N. J. A. Sloane

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff
Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000
This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022
a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003
a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005
Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005
For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006
Lengths of rows of triangle A182469;
Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013
Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014
Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014
A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016
Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016
a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016
a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016
a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...
From _Omar E. Pol_, Nov 30 2020: (Start)
For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.
Illustration of initial terms:
                              Diagram
   n   a(n)                         _
   1     1                        _|1|
   2     1                      _|1 _|
   3     2                    _|1  |1|
   4     1                  _|1   _| |
   5     2                _|1    |1 _|
   6     2              _|1     _| |1|
   7     2            _|1      |1  | |
   8     1          _|1       _|  _| |
   9     3        _|1        |1  |1 _|
  10     2      _|1         _|   | |1|
  11     2    _|1          |1   _| | |
  12     2   |1            |   |1  | |
...
a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).
Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 11.
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_o(n).
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.
N. J. A. Sloane, Transforms.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Eric Weisstein's World of Mathematics, Binomial Number and Odd Divisor Function.
Eric Weisstein's World of Mathematics, q-Polygamma Function.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000079, A000593, A010054 (char. func.), A038547 (positions of first appearances), A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.
Cf. A000203, A001620, A002162, A053866, A060831.
If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.
Dirichlet inverse is A327276.

a001227 = sum . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011

[NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019

for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
A001227 := proc(n) local a,d;
    a := 1 ;
    for d in ifactors(n)[2] do
        if op(1,d) > 2 then
            a := a*(op(2,d)+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 18 2015

f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 27 2004 *)
Table[Total[Mod[Divisors[n], 2]],{n,105}] (* Zak Seidov, Apr 16 2010 *)
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* Michael Somos, May 17 2013 *)
a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* Michael Somos, May 17 2013 *)
Count[Divisors[#],?OddQ]&/@Range[110] (* _Harvey P. Dale, Feb 15 2015 *)
(* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)
(* cl = current level, cs = current subparts count *)
a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]Hartmut F. W. Hoft, Dec 16 2016 *)
a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

{a(n) = sumdiv(n, d, d%2)}; /* Michael Somos, Oct 06 2007 */

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* Michael Somos, Oct 06 2007 */

a(n)=numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Mar 16 2011

a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013

a(n)=sumdivmult(n,d,d%2) \\ Charles R Greathouse IV, Aug 29 2013

from functools import reduce
from operator import mul
from sympy import factorint
def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # Chai Wah Wu, Mar 08 2021

def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])
[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
Comment from N. J. A. Sloane, Dec 02 2020: (Start)
By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function.  It is:
A(x) =  x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...
  = Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))
  = Sum_{k >= 1} x^k/(1-x^(2*k))
  = Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k)  [Ramanujan, 2nd notebook, p. 355.].
(This incorporates comments from Vladeta Jovovic, Oct 16 2002 and  Michael Somos, Oct 30 2005.) (End)
G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - Joerg Arndt, Nov 06 2010
a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005
Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.
a(n) = A113414(2*n). - N. J. A. Sloane, Jan 24 2006 (corrected Nov 10 2007)
a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, Apr 18 2006
Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007
a(n) = A000005(n) - A183063(n).
a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - Gary W. Adamson, Mar 15 2011
Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011
a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - Reinhard Zumkeller, May 01 2012
a(n) = 1 + A069283(n). - R. J. Mathar, Jun 18 2015
a(A002110(n)/2) = n, n >= 1. - Altug Alkan, Sep 29 2015
a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - Juri-Stepan Gerasimov, Jul 17 2016
Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - Omar E. Pol, Feb 15 2017
a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - Danny Rorabaugh, Oct 03 2017
L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - Michael Somos, Jun 01 2019
a(n) = A003056(n) - A238005(n). - Omar E. Pol, Sep 12 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023
a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - Ridouane Oudra, Apr 13 2023

A001190 Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, 127912, 293547, 676157, 1563372, 3626149, 8436379, 19680277, 46026618, 107890609, 253450711, 596572387, 1406818759, 3323236238, 7862958391, 18632325319, 44214569100, 105061603969

Offset: 0

N. J. A. Sloane

Also number of n-node binary rooted trees (every node has outdegree <= 2) where root has degree 0 (only for n=1) or 1.
a(n+1) is the number of rooted trees with n nodes where the outdegree of every node is <= 2, see example. These trees are obtained by removing the root of the trees in the comment above. - Joerg Arndt, Jun 29 2014
Number of interpretations of x^n (or number of ways to insert parentheses) when multiplication is commutative but not associative. E.g., a(4) = 2: x(x*x^2) and x^2*x^2. a(5) = 3: (x*x^2)x^2, x(x*x*x^2) and x(x^2*x^2). [If multiplication is non-commutative then the answer is A000108(n-1). - Jianing Song, Apr 29 2022]
Number of ways to place n stars in a single bound stable hierarchical multiple star system; i.e., taking only the configurations from A003214 where all stars are included in single outer parentheses. - Piet Hut, Nov 07 2003
Number of colorations of Kn (complete graph of order n) with n-1 colors such that no triangle is three-colored. Two edge-colorations C1 and C2 of G are isomorphic iff exists an automorphism f (isomorphism between G an G) such that: f sends same-colored edges of C1 on same-colored edges of C2 and f^(-1) sends same-colored edges of C2 on same-colored edges of C1. - Abraham Gutiérrez, Nov 12 2012
For n>1, a(n) is the number of (not necessarily distinct) unordered pairs of free unlabeled trees having a total of n nodes. See the first entry in formula section. - Geoffrey Critzer, Nov 09 2014
Named after the English mathematician Ivor Etherington (1908-1994) and the Scottish mathematician Joseph Wedderburn (1882-1948). - Amiram Eldar, May 29 2021

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + ...
From _Joerg Arndt_, Jun 29 2014: (Start)
The a(6+1) = 11 rooted trees with 6 nodes as described in the comment are:
:           level sequence       outdegrees (dots for zeros)
:     1:  [ 0 1 2 3 4 5 ]    [ 1 1 1 1 1 . ]
:  O--o--o--o--o--o
:
:     2:  [ 0 1 2 3 4 4 ]    [ 1 1 1 2 . . ]
:  O--o--o--o--o
:           .--o
:
:     3:  [ 0 1 2 3 4 3 ]    [ 1 1 2 1 . . ]
:  O--o--o--o--o
:        .--o
:
:     4:  [ 0 1 2 3 4 2 ]    [ 1 2 1 1 . . ]
:  O--o--o--o--o
:     .--o
:
:     5:  [ 0 1 2 3 4 1 ]    [ 2 1 1 1 . . ]
:  O--o--o--o--o
:  .--o
:
:     6:  [ 0 1 2 3 3 2 ]    [ 1 2 2 . . . ]
:  O--o--o--o
:        .--o
:     .--o
:
:     7:  [ 0 1 2 3 3 1 ]    [ 2 1 2 . . . ]
:  O--o--o--o
:        .--o
:  .--o
:
:     8:  [ 0 1 2 3 2 3 ]    [ 1 2 1 . 1 . ]
:  O--o--o--o
:     .--o--o
:
:     9:  [ 0 1 2 3 2 1 ]    [ 2 2 1 . . . ]
:  O--o--o--o
:     .--o
:  .--o
:
:    10:  [ 0 1 2 3 1 2 ]    [ 2 1 1 . 1 . ]
:  O--o--o--o
:  .--o--o
:
:    11:  [ 0 1 2 2 1 2 ]    [ 2 2 . . 1 . ]
:  O--o--o
:     .--o
:  .--o--o
:
(End)

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M. Konvalinka and S. Wagner, The shape of random tanglegrams, arXiv preprint arXiv:1512.01168 [math.CO], 2015.
A. Ledda, G. Achaz, T. Wiehe and L. Ferretti, Decomposing the site frequency spectrum: the impact of tree topology on neutrality tests, arXiv preprint arXiv:1510.06748 [q-bio.PE], 2015.
Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, Toric Richardson varieties of Catalan type and Wedderburn-Etherington numbers, arXiv:2105.12274 [math.AG], 2021.
F. Murtagh, Counting dendrograms: a survey, Discrete Applied Mathematics, 7 (1984), 191-199.
C. D. Olds, Problem 4277, Amer. Math. Monthly, 56 (1949), 697-699.
C. D. Olds (Proposer) and H. W. Becker (Discussion), Problem 4277, Amer. Math. Monthly 56 (1949), 697-699. [Annotated scanned copy]
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
David Serena and William J. Buchanan, Equivalence Classes Induced by Binary Tree Isomorphism -- Generating Functions, arXiv:2503.02663 [math.CO], 2025. See p. 5.
Chloe E. Shiff and Noah A. Rosenberg, Enumeration of rooted binary perfect phylogenies, arXiv:2410.14915 [q-bio.PE], 2024. See pp. 2, 5, 9.
F. Sievers, G. M. Hughes, and D. G. Higgins, Systematic Exploration of Guide-Tree Topology Effects for Small Protein Alignments, BMC Bioinformatics 2014, 15:338 (Mentions A001190).
J. H. M. Wedderburn, The functional equation g(x^2) = 2ax + [g(x)]^2, Ann. Math., 24 (1922-23), 121-140.
Eric Weisstein's World of Mathematics, Weakly Binary Tree.
Eric Weisstein's World of Mathematics, Strongly Binary Tree.
Wikipedia, Wedderburn-Etherington numbers.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for sequences related to parenthesizing

Cf. A000108, A001699, A002658, A003214, A006894, A006961, A088325.
Cf. A086317, A086318, A240943.
Cf. A292553, A292554, A292555, A292556.
Column k=2 of A292085 and of A299038.
Column k=1 of A319539 and of A319541.

A001190 := proc(n) option remember; local s,k; if n<=1 then RETURN(n); elif n <=3 then RETURN(1); else s := 0; if n mod 2 = 0 then s := A001190(n/2)*(A001190(n/2)+1)/2; for k from 1 to n/2-1 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); else for k from 1 to (n-1)/2 do s := s+A001190(k)*A001190(n-k); od; RETURN(s); fi; fi; end;
N := 40: G001190 := add(A001190(n)*x^n,n=0..N);
spec := [S,{S=Union(Z,Prod(Z,Set(S,card=2)))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
# alternative Maple program:
a:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
      (t-> t*(1-t)/2)(a(n/2)))+add(a(i)*a(n-i), i=1..n/2))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 28 2017

terms = 35; A[] = 0; Do[A[x] = x + (1/2)*(A[x]^2 + A[x^2]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Jul 22 2011, updated Jan 10 2018 *)
a[n_?OddQ] := a[n] = Sum[a[k]*a[n-k], {k, 1, (n-1)/2}]; a[n_?EvenQ] := a[n] = Sum[a[k]*a[n-k], {k, 1, n/2-1}] + (1/2)*a[n/2]*(1+a[n/2]); a[0]=0; a[1]=1; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 13 2012, after recurrence formula *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Nest[ 1 - Sqrt[1 - 2 x - (# /. x -> x^2)] &, 0, BitLength @ n], {x, 0, n}]]; (* Michael Somos, Apr 25 2013 *)

{a(n) = local(A, m); if( n<0, 0, m=1; A = O(x); while( m<=n, m*=2; A = 1 - sqrt(1 - 2*x - subst(A, x, x^2))); polcoeff(A, n))}; /* Michael Somos, Sep 06 2003 */

{a(n) = local(A); if( n<4, n>0, A = vector(n, i, 1); for( i=4, n, A[i] = sum( j=1, (i-1)\2, A[j] * A[i-j]) + if( i%2, 0, A[i/2] * (A[i/2] + 1)/2)); A[n])}; /* Michael Somos, Mar 25 2006 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A001190(n):
    if n <= 1: return n
    m = n//2 + n % 2
    return sum(A001190(i+1)*A001190(n-1-i) for i in range(m-1)) + (1 - n % 2)*A001190(m)*(A001190(m)+1)//2 # Chai Wah Wu, Jan 14 2022

G.f. satisfies A(x) = x + (1/2)*(A(x)^2 + A(x^2)) [de Bruijn and Klarner].
G.f. also satisfies A(x) = 1 - sqrt(1 - 2*x - A(x^2)). - Michael Somos, Sep 06 2003
a(2n-1) = a(1)a(2n-2) + a(2)a(2n-3) + ... + a(n-1)a(n), a(2n) = a(1)a(2n-1) + a(2)a(2n-2) + ... + a(n-1)a(n+1) + a(n)(a(n)+1)/2.
Given g.f. A(x), then B(x) = -1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = (u^2 + v)^2 + 2*(v^2 + w). - Michael Somos, Oct 22 2006
The radius of convergence of the g.f. is A240943 = 1/A086317 ~ 0.4026975... - Jean-François Alcover, Jul 28 2014, after Steven R. Finch.
a(n) ~ A086318 * A086317^(n-1) / n^(3/2). - Vaclav Kotesovec, Apr 19 2016

A001832 Number of labeled connected bipartite graphs on n nodes.

Original entry on oeis.org

1, 1, 3, 19, 195, 3031, 67263, 2086099, 89224635, 5254054111, 426609529863, 47982981969979, 7507894696005795, 1641072554263066471, 502596525992239961103, 216218525837808950623459, 130887167385831881114006475, 111653218763166828863141636911

Offset: 1

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..50
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]
Eric Weisstein's World of Mathematics, n-Colorable Graph
Eric Weisstein's World of Mathematics, n-Chromatic Graph

Row sums of A228861.
The unlabeled version is A005142.
Cf. A002031, A047863, A047864.

mx = 17; s = Sum[ Binomial[n, k] 2^(k (n - k)) x^n/n!, {n, 0, mx}, {k, 0, n}] ; Range[0, mx]! CoefficientList[ Series[ Log[s]/2, {x, 0, mx}], x] (* Geoffrey Critzer, May 10 2011 *)

seq(n)=Vec(serlaplace(log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2)) \\ Andrew Howroyd, Sep 26 2018

E.g.f.: log(A(x))/2 where A(x) is e.g.f. of A047863.

a(n) = A002031(n)/2, for n > 1. - Geoffrey Critzer, May 10 2011

More terms from Vladeta Jovovic, Apr 12 2003

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

Original entry on oeis.org

1, 1, 2, 7, 23, 115, 694, 5282, 46066, 456454, 4999004, 59916028, 778525516, 10897964660, 163461964024, 2615361578344, 44460982752488, 800296985768776, 15205638776753680, 304112757426239984, 6386367801916347184

Offset: 1

N. J. A. Sloane

			For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
Eric Weisstein's World of Mathematics, Rook Number.
Eric Weisstein's World of Mathematics, Rooks Problem.

Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903
P:=n->n!; # Gives A000142
G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)
rho:=n->R(n)/2; # Gives A000407, aerated
beta:=n->B(n)/2; # Gives A000902, doubled up
delta:=n->(D(n)-B(n))/2; # Gives A000900
unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up
alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

c[n_] := Floor[n/2]! 2^Floor[n/2];
r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
Array[a, 21] (* Jean-François Alcover, Apr 06 2011, after Matthias Engelhardt, further improved by Robert G. Wilson v *)

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e., factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions] and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal]. - Matthias Engelhardt, Apr 05 2000

For asymptotics see the Robinson paper.

More terms from David W. Wilson, Jul 13 2003

A000672 Number of 3-valent trees (= boron trees or binary trees) with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 6, 11, 18, 37, 66, 135, 265, 552, 1132, 2410, 5098, 11020, 23846, 52233, 114796, 254371, 565734, 1265579, 2841632, 6408674, 14502229, 32935002, 75021750, 171404424, 392658842, 901842517, 2076217086, 4790669518, 11077270335

Offset: 0

N. J. A. Sloane

This can be described in 2 ways: (a) Trees with n nodes of valency <= 3, for n = 0,1,2,3,... (b) Trees with t = 2n+2 nodes of valency either 1 or 3 (implying that there are n nodes of valency 3 - the boron atoms - and n+2 nodes of valency 1 - the hydrogen atoms), for t = 2,4,6,8,...

Essentially the same sequences arises from studying the number of unrooted, unlabeled binary tree topologies with n leaves (see A129860). - Steven Kelk, Jul 22 2016

			The 4 trees with 6 nodes are:
._._._._._. . ._._._._. . ._._._._. . ._._._.
. . . . . . . . | . . . . . . | . . . . | |
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 11*x^8 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 307.
P. J. Cameron, Oligomorphic Permutation Groups, Cambridge; see Fig. 2 p. 35.
A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).
Joseph Felsenstein, Inferring Phylogenies. Sinauer Associates, Inc., 2004, page 33. Note that at least the first two editions give an incorrect version of this sequence.
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 0..191 [b-file corrected by _N. J. A. Sloane_, Oct 04 2010]
Dominik Bendle, Janko Boehm, Yue Ren, and Benjamin Schröter, Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians, arXiv:2003.13752 [math.AG], 2020.
Nicolas Broutin and Philippe Flajolet, The distribution of height and diameter in random non-plane binary trees, Random Struct. Algorithms 41, No. 2, 215-252 (2012).
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183.
M. Chan, Tropical hyperelliptic curves, arXiv preprint arXiv:1110.0273 [math.CO], 2011.
Sergio Consoli, Jan Korst, Gijs Geleijnse, and Steffen Pauws, On the minimum quartet tree cost problem, arXiv:1807.00566 [cs.DM], 2018.
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Enumeration of constitutional isomers of polyenes, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
V. Fack, S. Lievens, and J. Van der Jeugt, On the diameter of the rotation graph of binary coupling trees, Discrete Math. 245 (2002), no. 1-3, 1--18. MR1887046 (2003i:05047).
Jean-François Fortin, Wen-Jie Ma, and Witold Skiba, Six-Point Conformal Blocks in the Snowflake Channel, arXiv:2004.02824 [hep-th], 2020.
Jean-François Fortin, Wen-Jie Ma, and Witold Skiba, Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond, arXiv:2006.13964 [hep-th], 2020.
Jean-François Fortin, Wen-Jie Ma, and Witold Skiba, All Global One- and Two-Dimensional Higher-Point Conformal Blocks, arXiv:2009.07674 [hep-th], 2020.
M. D. Hendy, C. H. C. Little, David Penny, Comparing trees with pendant vertices labelled, SIAM J. Appl. Math. 44 (5) (1984) Table 1
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, PhD Dissertation, University of South Carolina, 2012. - From _N. J. A. Sloane_, Dec 22 2012
R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599 discusses asymptotics.
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
Eric Weisstein's World of Mathematics, Trivalent Tree
Index entries for sequences related to trees

Column k = 3 of A144528.
A000672 = A000675 + A000673.
Cf. A001190(n+1) (rooted trees).
Cf. A052120, A000022, A000200, A000602, A129860.

(* c = A001190 *) c[n_?OddQ] := c[n] = Sum[c[k]*c[n-k], {k, 1, (n-1)/2}]; c[n_?EvenQ] := c[n] = (1/2)*c[n/2]*(c[n/2] + 1) + Sum[c[k]*c[n-k], {k, 1, n/2-1}]; c[0] = 0; c[1] = 1; b[x_] := If[IntegerQ[x], c[x+1], 0]; a[0] = a[1] = a[2] = 1; a[n_] := b[n/2] - (1/3)*(b[(n-1)/3]-1)*b[(n-1)/3]*(b[(n-1)/3] + 1) + 2*b[n] - b[n+1] - Sum[(1/2)*(b[i]-1)*b[i]*b[-2*i + n - 1], {i, 1, (n-2)/2}] + Sum[b[i]*Sum[b[j]*b[n-i-j-1], {j, i, (1/2)*(n-i-1)}], {i, 1, (n-1)/3}]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 19 2015 *)
n = 50; (* algorithm from Rains and Sloane *)
S2[f_,h_,x_] := f[h,x]^2/2 + f[h,x^2]/2;
S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S2[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + S3[T,h-1,z]z - S3[T,h-2,z]z - (T[h-1,z] - T[h-2,z]) (T[h-1,z]-1),z], n+1], {h,1,n/2}] + PadRight[{1,1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* Robert A. Russell, Sep 15 2018 *)
n = 60; c[n_?OddQ] := c[n] = Sum[c[k]*c[n-k], {k,1,(n-1)/2}];
c[n_?EvenQ] := c[n] = (1/2)*c[n/2]*(c[n/2] + 1) + Sum[c[k]*c[n-k],
  {k,1,n/2-1}]; c[0] = 0; c[1] = 1; (* as in program 1 above *)
gf[x_] := Sum[c[i+1] x^i, {i,0,n}]; (* g.f. for A001190(n+1) *)
ci[x_] := SymmetricGroupIndex[3, x] /. x[i_] -> gf[x^i];
CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 +
x ci[x], {x,0,n}]], x] (* Robert A. Russell, Jan 17 2023 *)

Rains and Sloane give a g.f.
a(0)=a(1)=a(2)=1, a(n) = 2*b(n+1) - b(n+2) + b((n+1)/2) - 2*C(1+b(n/3), 3) - Sum_{i=1..[(n-1)/2]} C(b(i), 2)*b(n-2*i) + Sum_{i=1..[n/3]} b(i) Sum_{j=i..[(n-i)/2]} b(j)*b(n-i-j), where b(x) = A001190(x+1) if x is an integer, otherwise 0 (Cyvin et al.) [Index of A001190 shifted by R. J. Mathar, Mar 08 2010]
a(n) = A000673(n) + A000675(n).
a(n) ~ c * d^n / n^(5/2), where d = A086317 = 2.4832535361726368585622885181... and c = 1.2551088797592580080398489829149157375... . - Vaclav Kotesovec, Apr 19 2016
G.f.: B(x) - cycle_index(S2,-B(x)) + x * cycle_index(S3,B(x)) = B(x) - (B(x)^2 - B(x^2)) / 2 + x * (B(x)^3 + 3*B(x) B(x^2) + 2*B(x^3)) / 6, where B(x) = 1 + x * cycle_index(S2,B(x)) = 1 + x * (B(x)^2 + B(x^2)) / 2 is the generating function for A001190(n+1). - Robert A. Russell, Jan 17 2023

A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 96, 32, 1, 1, 80, 640, 640, 80, 1, 1, 192, 3840, 10240, 3840, 192, 1, 1, 448, 21504, 143360, 143360, 21504, 448, 1, 1, 1024, 114688, 1835008, 4587520, 1835008, 114688, 1024, 1, 1, 2304, 589824, 22020096, 132120576, 132120576, 22020096, 589824, 2304, 1

Offset: 0

Emeric Deutsch, Aug 09 2005

Row sums yield A047863. T(2*n,n) = A111637(n). T(n,1) = A001787(n).

			T(2,1)=4 because we have B G, B--G, G B and G--B, where B (G) stands for a blue (green) node and -- denotes an edge.
Triangle starts:
  1;
  1,  1;
  1,  4,  1;
  1, 12, 12,  1;
  1, 32, 96, 32, 1;
  ...

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

S. R. Finch, Bipartite, k-colorable and k-colored graphs
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A047863, A111637, A001787.
Cf. A134530 (matrix log), A134531.
Cf. A000684, A011266, A038845, A140802, A224069 (matrix inverse).

T:=(n,k)->binomial(n,k)*2^(k*(n-k)): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

nn=6;f[x_,y_]:=Sum[Exp[x 2^n](y x)^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[f[x,y],{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Sep 07 2013 *)

T(n, k)=2^[k(n-k)]*C(n, k).
Matrix log yields triangle A134530, where A134530(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k). - Paul D. Hanna, Nov 11 2007
From Peter Bala, Aug 13 2012: (Start)
Let f(n) = n!*2^binomial(n,2) = A011266(n). Then T(n,k) = f(n)/(f(k)*f(n-k)).
E.g.f.: Sum_{n >= 0} exp(2^n*t*x)*x^n/n! = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + ....
O.g.f.: Sum_{n >= 0} x^n/(1-2^n*t*x)^(n+1) = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... O.g.f. for column k: 1/(1-2^k*x)^(k+1).
Recurrence equation: T(n,k) = 2^k*T(n-1,k) + 2^(n-k)*T(n-1,k-1).
Column k = 2: A038845. Column k = 3: A140802. Sum_{k = 0..n} k*T(n,k) = n*A000684(n). (End)
From Peter Bala, Apr 09 2013: (Start)
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function for this sequence is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 4*x + x^2)*z^2/(2!*2) + (1 + 12*x + 12*x^2 + x^3)*z^3/(3!*2^3) + .... Cf. Pascal's triangle A007318 with an e.g.f. of exp(z)*exp(x*z).
This is a generalized Riordan array (E(x), x) with respect to the sequence n!*2^C(n,2), as defined by Wang and Wang.
The n-th power of this triangle has a generating function E(z)^n*E(x*z). See A224069 for the inverse array (n = -1).
The n-th row is a log-concave sequence and hence unimodal.
The row polynomials satisfy the recurrence equation R(n+1,x) = 2^n*x*R(n,x/2) + R(n,2*x) with R(0,x) = 1, as well as R'(n,2*x) = n*2^(n-1)*R(n-1,x) (the ' denotes differentiation w.r.t. x). The row polynomials appear to have only real zeros.
Sum_{k = 0..n} (-1)^k*T(2*n+1,k) = 0;
Sum_{k = 0..n} (-1)^k*2^k*T(2*n,k) = 0;
Sum_{k = 0..n} 2^k*T(n,k) = A000684(n). (End)
T(n,k+1) = Product_{i=0..k} (T(n-i,1)/T(i+1,1)) for 0 <= k < n. - Werner Schulte, Nov 13 2018

A361950 Array read by antidiagonals: T(n,k) = n! * Sum_{s} 2^(Sum_{i=1..k-1} s(i)*s(i+1))/(Product_{i=1..k} s(i)!) where the sum is over all nonnegative compositions s of n into k parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 13, 26, 1, 0, 1, 5, 22, 81, 162, 1, 0, 1, 6, 33, 166, 721, 1442, 1, 0, 1, 7, 46, 287, 1726, 9153, 18306, 1, 0, 1, 8, 61, 450, 3309, 24814, 165313, 330626, 1, 0, 1, 9, 78, 661, 5650, 50975, 494902, 4244481, 8488962, 1, 0

Offset: 0

Andrew Howroyd, Mar 31 2023

T(n,k) corresponds to c(k,n) in the Klarner reference. This is an intermediate step in the computation of the number of labeled weakly graded (ranked) posets. The number of elements in the poset is n and the rank k.

			Array begins:
======================================================
n/k| 0 1      2       3        4        5        6 ...
---+--------------------------------------------------
0  | 1 1      1       1        1        1        1 ...
1  | 0 1      2       3        4        5        6 ...
2  | 0 1      6      13       22       33       46 ...
3  | 0 1     26      81      166      287      450 ...
4  | 0 1    162     721     1726     3309     5650 ...
5  | 0 1   1442    9153    24814    50975    91866 ...
6  | 0 1  18306  165313   494902  1058493  1957066 ...
7  | 0 1 330626 4244481 13729846 29885567 55363650 ...
  ...
T(3,2) = 26: the nonnegative integer compositions of 3 with 2 parts are (0,3), (1,2), (2,1), (3,0). These contribute, respectively 2^0*3!/(0!*3!) = 1, 2^2*3!/(1!*2!) = 12, 2^2*3!/(2!*1!) = 12, 2^0*3!/(0!*3!) = 1, so T(3,2) = 1 + 12 + 12 + 1 = 26.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.

Columns k=0..7 are A000007, A000012, A047863, A000684, A001827, A001828, A001829, A001830.
Rows 0..2 are A000012, A000027, A028872(n+1).
The unlabeled version is A361952.
Cf. A361951.

S(M)={matrix(#M, #M, i, j, sum(k=0, i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
C(n, m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
R(n)={Mat([Col(serlaplace(p)) | p<-C(n)])}
{ my(A=R(6)); for(i=1, #A, print(A[i,])) }

A000671 Number of boron trees with n nodes, i.e. n-node rooted trees with degree <= 3 at root and out-degree <= 2 elsewhere.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 14, 29, 60, 127, 275, 598, 1320, 2936, 6584, 14858, 33744, 76999, 176557, 406456, 939241, 2177573, 5064150, 11809632, 27610937, 64705623, 151966597, 357623905, 843176524, 1991439229, 4711115672, 11162025770, 26484061667, 62923251955

Offset: 0

N. J. A. Sloane

The subsequence of primes begins: 2, 7, 29, 127, 176557, 2177573, 151966597.

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 450).
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Enumeration of constitutional isomers of polyenes, J. Molec. Struct. (Theochem) 357, no. 3 (1995) 255-261.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
Index entries for sequences related to trees
Index entries for sequences related to rooted trees

Cf. A001190, A036657, A036658.

N := 40: t1 := G001190/x: G000671 := series(x*(1/3!)*(t1^3+3*subs(x=x^2,t1)*t1+2*subs(x=x^3,t1)), x, N); A000671 := n->coeff(G000671,x,n);
CI2 := proc(f) (1/2)*(f^2+subs(x=x^2,f)); end; CI3 := proc(f) (1/6)*(f^3+3*subs(x=x^2,f)*f+2*subs(x=x^3,f)); end;
N := 40: B0 := series(1 + x,x,N): G000671 := series(x*(CI3(B0) + CI3(G036656) + CI3(G036657) + CI2(B0)*(G036656 + G036657) + CI2(G036656)*(G036657 + B0) + CI2(G036657)*(B0 + G036656) + B0*G036656*G036657),x,N); A036658 := n->coeff(G036658,x,n);

terms = 32; (* B = g.f. for A001190 *) B[] = 0; Do[B[x] = x + (1/2)*(B[x]^2 + B[x^2]) + O[x]^terms // Normal, terms];
f[x_] = B[x]/x;
A[x_] = x*(1/3!)*(f[x]^3 + 3*f[x^2]*f[x] + 2*f[x^3]) + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, May 29 2012, from first g.f., updated Jan 10 2018 *)

G.f.: A(x) = x*(1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)), where f = G001190(x)/x, G001190 = g.f. for A001190.
a(n) = A001190(n) + A036657(n) + A036658(n).
Another g.f.: let B0(x) = 1+x, G036656(x) = g.f. for A036656, G036657(x) = g.f. for A036657.
Then g.f.: x*(cycle_index(S3, B0)+cycle_index(S3, G036656)+cycle_index(S3, G036657)+cycle_index(S2, B0)*(G036656+G036657)+cycle_index(S2, G036656)*(G036657+B0)+cycle_index(S2, G036657)*(B0+G036656)+B0*G036656*G036657), where cycle_index(Sk, f) means apply the cycle index for the symmetric group S_k to f(x).
E.g., cycle_index(S2, f) = (1/2!)*(f^2+subs(x=x^2, f), cycle_index(S3, f) = (1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)).

cp's OEIS Frontend

A001227 Number of odd divisors of n.

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A001190 Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).

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A001832 Number of labeled connected bipartite graphs on n nodes.

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A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

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A000672 Number of 3-valent trees (= boron trees or binary trees) with n nodes.

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A111636 Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

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