A107111 -id:A107111 - cp's OEIS frontend

A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

1, 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 4092553752, 37714212564, 350658882768, 3285490743987, 30989950019532, 294031964658430, 2804331954047160, 26870823304476690, 258548658860327880

Offset: 1

N. J. A. Sloane, Mira Bernstein

Apart from the initial 1, reversion of g.f. for A162395 (squares with signs): see A263843.

			G.f. = x*(1 + x + 4*x^2 + 23*x^3 + 156*x^4 + 1162*x^5 + 9192*x^6 + 75819*x^7 + ...).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..999 (first 101 terms from T. D. Noe)
M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Appl. Algebra, 191 (2004), 205-221.
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358 (column sums in Table 2).
Michael Drmota, Anna de Mier, Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (3). - N. J. A. Sloane, May 18 2014
Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144.
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486.
Loïc Foissy, Free quadri-algebras and dual quadri-algebras, arXiv:1504.06056 [math.CO], 2015.
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
Marco Kuhlmann, Tabulation of Noncrossing Acyclic Digraphs, arXiv:1504.04993 [cs.DS], 2015.
M. Kuhlmann, P. Jonsson, Parsing to Noncrossing Dependency Graphs, Transactions of the Association for Computational Linguistics, vol. 3, pp. 559-570, 2015.
Sara Madariaga, Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras, arXiv:1304.5184 [math.RA], 2013.
B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, arXiv:math/0609002 [math.QA], 2006-2007; J. Reine Angew. Math. 620 (2008), 105-164.
Gus Wiseman, The a(5) = 156 connected non-crossing graphs.
Gus Wiseman, Constructive Mathematica program for A007297.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Index entries for reversions of series

Cf. A162395, A000290. 4th row of A107111. Row sums of A089434.
See A263843 for a variant.
Cf. A000108 (non-crossing set partitions), A001006, A001187, A054726 (non-crossing graphs), A054921, A099947, A194560, A293510, A323818, A324167, A324169, A324173.

A007297:=proc(n) if n = 1 then 1 else add(binomial(3*n - 3, n + j)*binomial(j - 1, j - n + 1), j = n - 1 .. 2*n - 3)/(n - 1); fi; end;

CoefficientList[ InverseSeries[ Series[(x-x^2)/(1+x)^3, {x, 0, 20}], x], x] // Rest (* From Jean-François Alcover, May 19 2011, after PARI prog. *)
Table[Binomial[3n, 2n+1] Hypergeometric2F1[1-n, n, 2n+2, -1]/n, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^3+O(x^(n+2))),n+1)) /* Ralf Stephan */

Apart from initial term, g.f. is the series reversion of (x-x^2)/(1+x)^3 (A162395). See A263843. - Vladimir Kruchinin, Feb 08 2013
G.f.: (g-z)/z, where g=-1/3+(2/3)*sqrt(1+9z)*sin((1/3)*arcsin((2+27z+54z^2)/2/(1+9*z)^(3/2))). - Emeric Deutsch, Dec 02 2002
a(n) = (1/n)*Sum_{k=0..n} binomial(3n, n-k-1)*binomial(n+k-1, k). - Paul Barry, May 11 2005
a(n) = 4^(n-1)*(Gamma(3*n/2-1)/Gamma(n/2+1)/Gamma(n) -Gamma((3*n-1)/2)/ Gamma( (n+1)/2)/Gamma(n+1)). - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
a(n) = 4^n * binomial(3*n/2, n/2) / (9*n-6) - 4^(n-1) * binomial(3*(n-1)/2, (n-1)/2 ) / n. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
D-finite with recurrence: n*(n-1)*(3*n-4)*a(n) +36*(n-1)*a(n-1) -12*(3*n-8)*(3*n-1)*(3*n-7)*a(n-2)=0. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
a(n) = (1/n)*Sum_{k=0..n} C(3n, k)*C(2n-k-2, n-1). - Paul Barry, Sep 27 2005
a(n) ~ (2-sqrt(3)) * 6^n * 3^(n/2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
a(n) = binomial(3*n,2*n+1)*hypergeom([1-n,n], [2*n+2], -1)/n. - Vladimir Reshetnikov, Oct 25 2015
a(n) = 2*A078531(n) - A085614(n+1). - Vladimir Reshetnikov, Apr 24 2016

Better description from Philippe Flajolet, Apr 20 2000
More terms from James Sellers, Aug 21 2000
Definition revised and initial a(1)=1 added by N. J. A. Sloane, Nov 05 2015 at the suggestion of Axel Boldt. Some of the formulas may now need to be adjusted slightly.

A143689 a(n) = (3*n^2 - n + 2)/2.

Original entry on oeis.org

1, 2, 6, 13, 23, 36, 52, 71, 93, 118, 146, 177, 211, 248, 288, 331, 377, 426, 478, 533, 591, 652, 716, 783, 853, 926, 1002, 1081, 1163, 1248, 1336, 1427, 1521, 1618, 1718, 1821, 1927, 2036, 2148, 2263, 2381, 2502, 2626, 2753, 2883, 3016, 3152, 3291

Offset: 0

Gary W. Adamson, Aug 29 2008

Equals left border of triangle A033292.
Equals binomial transform of [1, 1, 3, 0, 0, 0, ...].
A242357(a(n)) = 1. - Reinhard Zumkeller, May 11 2014
These might be called "trisected pentagonal numbers": A figurate pentagonal number is composed of three triangles, of which the central one is the largest, and the removal of the triangular frame (3*n) of the central triangle trisects the figure. This is reflected in the formula a(n) = A000326(n+1) - 3*n. See illustration in links. - John Elias, May 27 2022

Michael De Vlieger, Table of n, a(n) for n = 0..10000
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
John Elias, Trisected Pentagonal Numbers
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A000326(n+1) - 3n. Third column of A107111.

Cf. A033292, A104249.

a143689 n = n*(3*n-1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014

Table[(3n^2-n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,6},50] (* Harvey P. Dale, May 05 2014 *)

makelist(binomial(n, 2) + n^2 + 1, n, 0, 100); /* Franck Maminirina Ramaharo, Mar 01 2018 */

a(n)=(3*n^2-n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000326(n+1) - 3*n. (A000326 are the pentagonal numbers.)
a(n) = (3*n^2 - n + 2)/2 = A027599(n+1)/2. - R. J. Mathar, Sep 03 2008
a(n) = a(n-1) + 3*n - 2 (with a(0)=1). - Vincenzo Librandi, Nov 25 2010
a(n) = 2*a(n-1) - a(n-2) + 3.
O.g.f.: (1-x+3*x^2)/((1-x)^3). - Eric Werley, Jun 27 2011
a(n) = A104249(-n). - Bruno Berselli, Jul 08 2015
a(n) = binomial(n,2) + n^2 + 1 = A152947(n+1) + A000290(n). - Franck Maminirina Ramaharo, Mar 01 2018
E.g.f.: exp(x)*(2 + 2*x + 3*x^2)/2. - Stefano Spezia, Apr 19 2025

Index of A000326 in definition, formula and example corrected by R. J. Mathar, Sep 03 2008

A365754 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x)^4 ).

Original entry on oeis.org

1, 5, 36, 305, 2833, 27916, 286632, 3033513, 32858595, 362515725, 4059475368, 46021411644, 527163783916, 6092053249160, 70939443268112, 831558454663449, 9804617762941095, 116201796106426543, 1383557994261012100, 16541672701743657545, 198510770031798279825

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Cf. A000108, A006318, A107111, A263843, A365755.

Cf. A365752.

a(n) = sum(k=0, n, binomial(n+k, k)*binomial(4*(n+1), n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(4*(n+1),n-k).

a(n) = (1/(n+1)) * [x^n] ( (1+x)^4 / (1-x) )^(n+1). - Seiichi Manyama, Feb 17 2024

A365755 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x)^5 ).

Original entry on oeis.org

1, 6, 52, 530, 5919, 70098, 864784, 10994490, 143042020, 1895316632, 25487708844, 346976558318, 4772478619146, 66222166440780, 925880434336320, 13030945427540170, 184467676431001644, 2624828100099166536, 37521220349342729680, 538573138719587026440

Offset: 0

Seiichi Manyama, Sep 18 2023

nonn

Cf. A000108, A006318, A107111, A263843, A365754.

Cf. A365753.

a(n) = sum(k=0, n, binomial(n+k, k)*binomial(5*(n+1), n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(5*(n+1),n-k).

A345462 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) read by rows: number of distinct permutations after k steps of the "first transposition" algorithm.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 13, 4, 1, 120, 67, 23, 5, 1, 720, 411, 146, 36, 6, 1, 5040, 2921, 1067, 272, 52, 7, 1, 40320, 23633, 8800, 2311, 456, 71, 8, 1, 362880, 214551, 81055, 21723, 4419, 709, 93, 9, 1, 3628800, 2160343, 825382, 224650, 46654, 7720, 1042, 118, 10, 1

Offset: 1

Olivier Gérard, Jun 20 2021

The first transposition algorithm is: if the permutation is sorted, then exit; otherwise, exchange the first unsorted letter with the letter currently at its index. Repeat.
At each step at least 1 letter (possibly 2) is sorted.
If one counts the steps necessary to reach the identity, this gives the Stirling numbers of the first kind (reversed).

			Triangle begins:
      1;
      2,     1;
      6,     3,    1;
     24,    13,    4,    1;
    120,    67,   23,    5,   1;
    720,   411,  146,   36,   6,  1;
   5040,  2921, 1067,  272,  52,  7, 1;
  40320, 23633, 8800, 2311, 456, 71, 8, 1;
  ...

D. E. Knuth, The Art of Computer Programming, Vol. 3 / Sorting and Searching, Addison-Wesley, 1973.

Alois P. Heinz, Rows n = 1..150, flattened

Cf. A321352, A345461 (same idea for other sorting algorithms).
Cf. A180191 (second column, k=1).
Cf. A107111 a triangle with some common parts.
Cf. A143689 (diagonal T(n,n-3)).
Cf. A000142, A000166, A000255, A000274, A000313, A010027, A123513.

b:= proc(n, k) option remember; (k+1)!*
      binomial(n, k)*add((-1)^i/i!, i=0..k+1)/n
    end:
T:= proc(n, k) option remember;
     `if`(k=0, n!, T(n, k-1)-b(n, n-k+1))
    end:
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Aug 11 2021

b[n_, k_] := b[n, k] = (k+1)!*Binomial[n, k]*Sum[(-1)^i/i!, {i, 0, k+1}]/n;
T[n_, k_] := T[n, k] = If[k == 0, n!, T[n, k-1] - b[n, n-k+1]];
Table[Table[T[n, k], {k, 0, n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

T(n,0) = n!; T(n,n-3) = (3*(n-1)^2 - n + 3)/2.
From Alois P. Heinz, Aug 11 2021: (Start)
T(n,k) = T(n,k-1) - A010027(n,n-k) for k >= 1.
T(n,k) - T(n,k+1) = A123513(n,k).
T(n,0) - T(n,1) = A000255(n-1) for n >= 2.
T(n,1) - T(n,2) = A000166(n) for n >= 3.
T(n,2) - T(n,3) = A000274(n) for n >= 4.
T(n,3) - T(n,4) = A000313(n) for n >= 5. (End)

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A107112 Row sums of A107111, viewed as a number triangle.

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A107113 Diagonal sums of A107111, viewed as a number triangle.

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A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

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A143689 a(n) = (3*n^2 - n + 2)/2.

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A365754 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x)^4 ).

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A365755 Expansion of (1/x) * Series_Reversion( x*(1-x)/(1+x)^5 ).

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A345462 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) read by rows: number of distinct permutations after k steps of the "first transposition" algorithm.

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A366012 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k).

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