A263843 -id:A263843 - 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.

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

A162395 a(n) = -(-1)^n * n^2.

Original entry on oeis.org

1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, -144, 169, -196, 225, -256, 289, -324, 361, -400, 441, -484, 529, -576, 625, -676, 729, -784, 841, -900, 961, -1024, 1089, -1156, 1225, -1296, 1369, -1444, 1521, -1600, 1681, -1764, 1849, -1936, 2025, -2116, 2209, -2304, 2401, -2500

Offset: 1

Michael Somos, Jul 02 2009

This sequence is the denominator of (Pi^2)/12 = 1/1-1/4+1/9-1/16+1/25-1/36+... - Mohammad K. Azarian, Dec 29 2011

Also, circulant determinant of [1,2,...,n,n-1,...,1], i.e., determinant of the (2n-1) X (2n-1) matrix which has this as first row (and also first column), where row k+1 is obtained by cyclically shifting row k one place to the left. - M. F. Hasler, Dec 17 2016

			G.f. = x - 4*x^2 + 9*x^3 - 16*x^4 + 25*x^5 - 36*x^6 + 49*x^7 - 64*x^8 + 81*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A000290, A072691.

For the reversion of this sequence see A263843 (and also A007297).

[(-1)^(n+1) * n^2: n in [1..60]]; // Vincenzo Librandi, Feb 15 2013

Table[(-1)^(n+1) * n^2, {n, 60}] (* Vincenzo Librandi, Feb 15 2013 *)

PARI
```
{a(n) = -(-1)^n * n^2};
```

Euler transform of length 2 sequence [-4, 3].
a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = (p^2)^e if p>2.
G.f.: x * (1 - x) / (1 + x)^3.
E.g.f.: exp(-x) * (x - x^2).
a(n) = a(-n) = -(-1)^n * A000290(n) for all n in Z.
Sum_{n>=1} 1/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Dec 10 2022
Dirichlet g.f.: zeta(s-2)*(1-2^(3-s)) = DirichletEta(s-2). - Amiram Eldar, Jan 07 2023

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).

A365816 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).

Original entry on oeis.org

0, 1, 6, 57, 658, 8442, 115692, 1658505, 24565530, 372999198, 5774883348, 90821581578, 1446901409268, 23301338376916, 378711707274072, 6203898306232233, 102329366764727658, 1698047225583890550, 28327664136201303300, 474821679792884860590, 7992739387298462213340

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Reversion of g.f. for hexagonal numbers (with signs).

Eric Weisstein's World of Mathematics, Hexagonal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A000384, A064087, A113207, A179848, A263843, A365817, A365818.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^3/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^3, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * 3^k for n > 0.

a(n) ~ 6^(3*n + 1/2) / (sqrt((481 + 133*sqrt(13))*Pi) * n^(3/2) * (13*sqrt(13) - 35)^n). - Vaclav Kotesovec, Sep 26 2023

A365817 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 4 * A(x)).

Original entry on oeis.org

0, 1, 7, 80, 1119, 17437, 290532, 5066364, 91311055, 1687341227, 31797227631, 608727899936, 11805599569092, 231454163924700, 4579765707561240, 91340133073920420, 1834295500622405295, 37059418988408887015, 752741444501505866325, 15362331852042084534240, 314860558967057266779495

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Reversion of g.f. for heptagonal numbers (with signs).

Eric Weisstein's World of Mathematics, Heptagonal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A000566, A064088, A113207, A179848, A263843, A365816, A365818.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^3/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 x)/(1 + x)^3, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] 4^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * 4^k for n > 0.

a(n) ~ 5 * (81 + 21*sqrt(21))^n / (sqrt((427 + 93*sqrt(21))*Pi) * n^(3/2) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Sep 26 2023

A365818 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 5 * A(x)).

Original entry on oeis.org

0, 1, 8, 107, 1760, 32298, 634128, 13034247, 276943568, 6033834950, 134069957840, 3026476515790, 69213144181888, 1600157697995092, 37337615574348960, 878166685063548639, 20797051344280763184, 495509950454603339310, 11869278747340342255440, 285669061791469915886250, 6904850429493240677872320

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Reversion of g.f. for octagonal numbers (with signs).

Eric Weisstein's World of Mathematics, Octagonal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A000567, A064089, A113207, A179848, A263843, A365816, A365817.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^3/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^3, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] 5^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * 5^k for n > 0.

a(n) ~ 3^(3/2) * 2^(n - 1/2) * (154 + 31*sqrt(31))^n / (sqrt((2821 + 506*sqrt(31))*Pi) * n^(3/2) * 5^(2*n)). - Vaclav Kotesovec, Sep 26 2023

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

Original entry on oeis.org

1, 2, 13, 156, 2833, 70098, 2214280, 85464984, 3906724321, 206648387550, 12425282899588, 837384222603448, 62539219710804627, 5127758187193514824, 457986530357734020432, 44263628968974498793648, 4602969726808566383149761, 512486177498084438210961270, 60827938291895363867587959628

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Main diagonal of A107111.

Cf. A000108, A006318, A062992, A135861, A263843, A365754, A365755.

Table[1/(n + 1) Sum[Binomial[n + k, n] Binomial[n (n + 1), n - k], {k, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 18}]

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - x) / (1 + x)^n ).

a(n) ~ exp(n + 3/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Sep 26 2023

A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3n,n-k-1) (n-3)^k.

Original entry on oeis.org

1, 2, 12, 156, 3507, 115692, 5066364, 276943568, 18152243967, 1387267590540, 121106707350928, 11889022355301672, 1296359140925188212, 155440199716271334648, 20327081449263918542412, 2879054747404226046119448, 439060192463001381367975215, 71727764882350305085962745740

Offset: 1

Ilya Gutkovskiy, Oct 04 2023

nonn

a(n) is the coefficient of x^n in expansion of series reversion of g.f. for n-gonal numbers (with signs).

Cf. A001764, A060354, A113207, A179848, A263843, A323208, A365816, A365817, A365818, A366204.

Unprotect[Power]; 0^0 = 1; Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] (n - 3)^k, {k, 0, n - 1}], {n, 1, 18}]
Table[Binomial[3 n, n - 1] Hypergeometric2F1[1 - n, n, 2 (n + 1), 3 - n]/n, {n, 1, 18}]
Table[SeriesCoefficient[InverseSeries[Series[x (1 - (n - 3) x)/(1 + x)^3, {x, 0, n}], x], {x, 0, n}], {n, 1, 18}]

a(n) = [x^n] Series_Reversion( x * (1 - (n - 3) * x) / (1 + x)^3 ).

A291536 Expansion of the series reversion of x(1 + 4x + x^2)/(1 - x)^4.

Original entry on oeis.org

1, -8, 101, -1544, 26190, -474144, 8975229, -175492664, 3516970490, -71858843264, 1491301438354, -31349284476496, 666133734882748, -14284509655611840, 308734263333717021, -6718525508918998872, 147085140049822666626, -3237191565662618280384, 71584853778205231503750

Offset: 1

Ilya Gutkovskiy, Aug 25 2017

sign

Reversion of g.f. for the cubes (A000578).

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Series Reversion
Index entries for reversions of series

Cf. A000578, A007297, A263843.

Rest[CoefficientList[InverseSeries[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 19}], x], x]]

G.f. A(x) satisfies: A(x)*(1 + 4*A(x) + A(x)^2)/(1 - A(x))^4 = x.
a(n) ~ -(-1)^n * (5*sqrt(6) - 12) * 2^(3*n-2) * 3^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 11 2017
D-finite with recurrence 7*n*(n-1)*(n+1)*a(n) +4*n*(n-1)*(58*n-83)*a(n-1) -36*(n-1)*(16*n^2-80*n+155)*a(n-2) +432*(-96*n^3+720*n^2-1794*n+1495)*a(n-3) +6912*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A162395 a(n) = -(-1)^n * n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365816 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A365817 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 4 * A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A365818 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 5 * A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Crossrefs

A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3n,n-k-1) (n-3)^k.

A291536 Expansion of the series reversion of x(1 + 4x + x^2)/(1 - x)^4.