A000699 -id:A000699 - cp's OEIS frontend

A036987 Fredholm-Rueppel sequence.

1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Binary representation of the Kempner-Mahler number Sum_{k>=0} 1/2^(2^k) = A007404.
Also a(n) = A(n) mod 2 where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363. - Philippe Deléham, May 26 2007
a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence (A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. Zizka, Feb 12 2008
a(n-1), n >= 1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: Product_{j>=1} (1 + a(j-1)*x^j) = 1 + Sum_{k>=1} x^k = 1/(1-x). The product is therefore Product_{l>=1} (1 + x^(2^l)). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. - Wolfdieter Lang, Mar 05 2009
a(n) is also the number of orbits of length n for the map x -> 1-cx^2 on [-1,1] at the Feigenbaum critical value c=1.401155... . - Thomas Ward, Apr 08 2009
A054525 (Mobius transform) * A001511 = A036987 = A047999^(-1) * A001511 = the inverse of Sierpiński's gasket * the ruler sequence. - Gary W. Adamson, Oct 26 2009 [Of course this is only vaguely correct depending on how the fuzzy indexing in these formulas is made concrete. - R. J. Mathar, Jun 20 2014]
Characteristic function of A000225. - Reinhard Zumkeller, Mar 06 2012
Also parity of the Catalan numbers A000108. - Omar E. Pol, Jan 17 2012
For n >= 2, also the largest exponent k >= 0 such that n^k in binary notation does not contain both 0 and 1. Unlike for the decimal version of this sequence, A062518, where the terms are only conjectural, for this sequence the values of a(n) can be proved to be the characteristic function of A000225, as follows: n^k will contain both 0 and 1 unless n^k = 2^r-1 for some r. But this is a special case of Catalan's equation x^p = y^q-1, which was proved by Preda Mihăilescu to have no nontrivial solution except 2^3 = 3^2 - 1. - Christopher J. Smyth, Aug 22 2014
Image, under the coding a,b -> 1; c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cb, c -> cc. - Jeffrey Shallit, May 14 2016
Number of nonisomorphic Boolean algebras of order n+1. - Jianing Song, Jan 23 2020

			G.f. = 1 + x + x^3 + x^7 + x^15 + x^31 + x^63 + x^127 + x^255 + x^511 + ...
a(7) = 1 since 7 = 2^3 - 1, while a(10) = 0 since 10 is not of the form 2^k - 1 for any integer k.

Antti Karttunen, Table of n, a(n) for n = 0..65537
D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Mats Granvik, Mathematica program for computing Fredholm Rueppel sequences
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.
Preda Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. doi:10.1515/crll.2004.048. MR 2076124.
H. Niederreiter and M. Vielhaber, Tree Complexity and a Doubly Exponential Gap between Structured and Random Sequences, J. Complexity, 12 (1996), 187-198.
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. [From _Thomas Ward_, Apr 08 2009]
Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 8.
E. Sheppard, net.math post (1985)
Stephen Wolfram, [Page 1092] A New Kind of Science | Online.
Index entries for characteristic functions

Cf. A007404, A043545, A062518, A078885, A078585, A078886, A078887, A078888, A078889, A078890.
The first row of A073346. Occurs for first time in A073202 as row 6 (and again as row 8).
Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.
If interpreted with offset=1 instead of 0 (i.e., a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.
This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).
Cf. A054525, A047999. - Gary W. Adamson, Oct 26 2009
Cf. A043529, A127802.

a036987 n = ibp (n+1) where
   ibp 1 = 1
   ibp n = if r > 0 then 0 else ibp n' where (n',r) = divMod n 2
a036987_list = 1 : f [0,1] where f (x:y:xs) = y : f (x:xs ++ [x,x+y])
-- Same list generator function as for a091090_list, cf. A091090.
-- Reinhard Zumkeller, May 19 2015, Apr 13 2013, Mar 13 2013

A036987:= n-> `if`(2^ilog2(n+1) = n+1, 1, 0):
seq(A036987(n), n=0..128);

RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]
(* Recurrence: *)
t[n_, 1] = 1; t[1, k_] = 1;
t[n_, k_] := t[n, k] =
  If[n < k, If[n > 1 && k > 1, -Sum[t[k - i, n], {i, 1, n - 1}], 0],
   If[n > 1 && k > 1, Sum[t[n - i, k], {i, 1, k - 1}], 0]];
Table[t[n, k], {k, n, n}, {n, 104}]
(* Mats Granvik, Jun 03 2011 *)
mb2d[n_]:=1 - Module[{n2 = IntegerDigits[n, 2]}, Max[n2] - Min[n2]]; Array[mb2d, 120, 0] (* Vincenzo Librandi, Jul 19 2019 *)
Table[PadRight[{1},2^k,0],{k,0,7}]//Flatten (* Harvey P. Dale, Apr 23 2022 *)

{a(n) =( n++) == 2^valuation(n, 2)}; /* Michael Somos, Aug 25 2003 */

a(n) = !bitand(n, n+1); \\ Ruud H.G. van Tol, Apr 05 2023

from sympy import catalan
def a(n): return catalan(n)%2 # Indranil Ghosh, May 25 2017

def A036987(n): return int(not(n&(n+1))) # Chai Wah Wu, Jul 06 2022

1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.
a(n) = a(floor(n/2)) * (n mod 2) for n>0 with a(0)=1. - Reinhard Zumkeller, Aug 02 2002 [Corrected by Mikhail Kurkov, Jul 16 2019]
Sum_{n>=0} 1/10^(2^n) = 0.110100010000000100000000000000010...
1 if n=0, floor(log_2(n+1)) - floor(log_2(n)) otherwise. G.f.: (1/x) * Sum_{k>=0} x^(2^k) = Sum_{k>=0} x^(2^k-1). - Ralf Stephan, Apr 28 2003
a(n) = 1 - A043545(n). - Michael Somos, Aug 25 2003
a(n) = -Sum_{d|n+1} mu(2*d). - Benoit Cloitre, Oct 24 2003
Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).
a(n) = A000108(n) mod 2 = A001405(n) mod 2. - Paul Barry, Nov 22 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{j=0..k} binomial(k, 2^j-1). - Paul Barry, Jun 01 2006
A000523(n+1) = Sum_{k=1..n} a(k). - Mitch Harris, Jul 22 2011
a(n) = A209229(n+1). - Reinhard Zumkeller, Mar 07 2012
a(n) = Sum_{k=1..n} A191898(n,k)*cos(Pi*(n-1)*(k-1))/n; (conjecture). - Mats Granvik, Mar 04 2013
a(n) = A000035(A000108(n)). - Omar E. Pol, Aug 06 2013
a(n) = 1 iff n=2^k-1 for some k, 0 otherwise. - M. F. Hasler, Jun 20 2014
a(n) = ceiling(log_2(n+2)) - ceiling(log_2(n+1)). - Gionata Neri, Sep 06 2015
From John M. Campbell, Jul 21 2016: (Start)
a(n) = (A000168(n-1) mod 2).
a(n) = (A000531(n+1) mod 2).
a(n) = (A000699(n+1) mod 2).
a(n) = (A000891(n) mod 2).
a(n) = (A000913(n-1) mod 2), for n>1.
a(n) = (A000917(n-1) mod 2), for n>0.
a(n) = (A001142(n) mod 2).
a(n) = (A001246(n) mod 2).
a(n) = (A001246(n) mod 4).
a(n) = (A002057(n-2) mod 2), for n>1.
a(n) = (A002430(n+1) mod 2). (End)
a(n) = 2 - A043529(n). - Antti Karttunen, Nov 19 2017
a(n) = floor(1+log(n+1)/log(2)) - floor(log(2n+1)/log(2)). - Adriano Caroli, Sep 22 2019
This is also the decimal expansion of -Sum_{k>=1} mu(2*k)/(10^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

Edited by M. F. Hasler, Jun 20 2014

A000698 A problem of configurations: a(0) = 1; for n>0, a(n) = (2n-1)!! - Sum_{k=1..n-1} (2k-1)!! a(n-k). Also the number of shellings of an n-cube, divided by 2^n n!.

Original entry on oeis.org

1, 1, 2, 10, 74, 706, 8162, 110410, 1708394, 29752066, 576037442, 12277827850, 285764591114, 7213364729026, 196316804255522, 5731249477826890, 178676789473121834, 5925085744543837186, 208256802758892355202, 7734158085942678174730

Offset: 0

N. J. A. Sloane, Richard Ehrenborg, Gabor Hetyei

Also number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams. [Corrected by Charles R Greathouse IV, Jan 24 2014][Clarified by Robert Coquereaux, Sep 14 2014]
a(n+1) is the moment of order 2*n for the probability density function rho(x) = (1/sqrt(2*Pi))*exp(x^2/2)/[(u(x))^2+Pi/2], with u(x) = Integral_{t=0..x} exp(t*t/2) dt, on the real interval -infinity..infinity. - Groux Roland, Jan 13 2009
Starting (1, 2, 10, 74, ...) = INVERTi transform of A001147: (1, 3, 15, 105, ...). - Gary W. Adamson, Oct 21 2009
The Cvitanovic et al. paper relates this sequence to A005411 and A005413. - Robert Munafo, Jan 24 2010
Hankel transform of a(n+1) is A168467. - Paul Barry, Nov 26 2009
a(n) = number of labeled Dyck (n-1)-paths (A000108) in which each vertex that terminates an upstep is labeled with an integer i in [0,h], where h is the height of the vertex . For example UDUD contributes 4 labeled paths--0D0D, 0D1D, 1D0D, 1D1D where upsteps are replaced by their labels--and UUDD contributes 6 labeled paths to a(3)=10. The Deléham (Mar 24 2007) formula below counts these labeled paths by number of "0" labels. - David Callan, Aug 23 2011
a(n) is the number of indecomposable perfect matchings on [2n]. A perfect matching on [2n] is decomposable if a nonempty subset of the edges forms a perfect matching on [2k] for some kDavid Callan, Nov 29 2012
From Robert Coquereaux, Sep 12 2014: (Start)
QED diagrams are graphs with two kinds of edges (lines): a (non-oriented), f (oriented), and only one kind of (internal) vertex: aff. They may have internal and external (i.e., pendant) lines. The order is the number of (internal) vertices. Vanishing diagrams: QED diagrams containing loops of type f with an odd number of vertices are set to 0 (Furry theorem). Proper diagrams: connected QED diagrams that remain connected when an arbitrary internal line is cut.
The number of Feynman diagrams of order 2n for the electron propagator (2-point function of QED), vanishing or not, proper or not, of order 2n, starting from n = 0, is given by 1, 2, 10, 74, 706, 8162, ..., i.e., this sequence A000698, with the first term (equal to 1) dropped. Call Sf the associated g.f.
The number of non-vanishing Feynman diagrams, for the same 2-point function, is given by 1, 1, 4, 25, 208, 2146, ..., i.e., by the sequence A005411, with a first term of order 0, equal to 1, added. Call S the associated g.f.
If one does not remove the vanishing diagram, but, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams (vanishing and non-vanishing) for the self-energy function of QED, 0, 1, 3, 21, 207, 2529, ..., i.e., the sequence A115974 with a first term of order 0, equal to 0, added. A115974 is twice A167872. Call Sigmaf the associated g.f.
If one removes the vanishing diagrams and, at the same time, considers only those graphs that are proper, one obtains the Feynman diagrams for the self-energy function of QED given by 0, 1, 3, 18, 153, 1638, ..., i.e., by the sequence A005412, with a first term of order 0, equal to 0, added. Call Sigma the associated g.f.
Then Sf = 1/(1-Sigmaf) and S = 1/(1-Sigma). (End)
For n>0 sum over all Dyck paths of semilength n-1 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 22 2015
Also, counts certain isomorphism classes of closed normal linear lambda terms. [N. Zeilberger, 2015]. - N. J. A. Sloane, Sep 18 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
For n >= 2, a(n) is the number of coalescent histories for a pair consisting of a matching lodgepole gene tree and species tree with 2n-1 leaves. - Noah A Rosenberg, Jun 21 2022

			G.f. = 1 + x + 2*x^2 + 10*x^3 + 74*x^4 + 706*x^5 + 8162*x^6 + 110410*x^7 + ...

Dubois C., Giorgetti A., Genestier R. (2016) Tests and Proofs for Enumerative Combinatorics. In: Aichernig B., Furia C. (eds) Tests and Proofs. TAP 2016. Lecture Notes in Computer Science, vol 9762. Springer.
R. W. Robinson, Counting irreducible Feynman diagrams exactly and asymptotically, Abstracts Amer. Math. Soc., 2002, #975-05-270.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Arques and J.-F. Beraud, Rooted maps on orientable surfaces, Riccati's equation and continued fractions, Discrete Math., 215 (2000), 1-12.
F. Battaglia and T. F. George, A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams, J. Math. Chem. 2 (1988) 241-247.
S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, arXiv preprint arXiv:1111.4710 [math.CO], 2011.
S. Birdsong and G. Hetyei, A Gray Code for the Shelling Types of the Boundary of a Hypercube, Discrete Math. 313 (2013), no. 3, 258-268. MR2995390.
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts)
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011, Lemma 3.2.
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Disc. Appl. Math. 161 (2013) 1378-1394
J. Courtiel, K. Yeats, and N. Zeilberger, Connected chord diagrams and bridgeless maps, arXiv:1661.04611, Proposition 13.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949, (eq 3.34 and fig 2b). DOI:10.1103/PhysRevD.18.1939
M. A. Deryagina and A. D. Mednykh, On the enumeration of circular maps with given number of edges, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.
F. Disanto and N. A. Rosenberg, Coalescent histories for lodgepole species trees, J. Comput. Biol. 22 (2015), 918-929.
S. Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv:1504.06904 [math.SP], 2015.
G. Edgar, Transseries for beginners, arXiv:0801.4877v5 [math.RA], 2008-2009.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.
R. J. Mathar, Table of Feynman Diagrams of the Interacting Fermion Green's Function, Int. J. Quant. Chem. 107 (2007) 1975. Also on arXiv, arXiv:physics/0512022 [physics.atom-ph], 2015-2016.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section VII.
L. G. Molinari, Hedin's equations and enumeration of Feynman diagrams, Phys. Rev. B, 71 (2005), 113102.
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J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.
J. Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25. [Annotated, corrected, scanned copy]
Wikipedia, Feynman diagram
Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030, March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.
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Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A004208, A000165, A001147, A005412, row sums of A053979, A005411, A115974, A167872.
Column k=1 of A258219, A258222.
Row sums of A322398.

A006882 := proc(n) option remember; if n <= 1 then 1 else n*procname(n-2); fi; end;
A000698:=proc(n) option remember; global df; local k; if n=0 then RETURN(1); fi; A006882(2*n-1) - add(A006882(2*k-1)*A000698(n-k),k=1..n-1); end;
A000698 := proc(n::integer) local resul,fac,pows,c,c1,p,i ; if n = 0 then RETURN(1) ; else pows := combinat[partition](n) ; resul := 0 ; for p from 1 to nops(pows) do c := combinat[permute](op(p,pows)) ; c1 := op(1,c) ; fac := nops(c) ; for i from 1 to nops(c1) do fac := fac*doublefactorial(2*op(i,c1)-1) ; od ; resul := resul-(-1)^nops(c1)*fac ; od : fi ; RETURN(resul) ; end; # R. J. Mathar, Apr 24 2006
# alternative Maple program:
b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> `if`(n=0, 1, b(2*n-2, 0, false)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 23 2015
a_list := proc(len) local n, A; if len=1 then return [1] fi: A := Array(-1..len-2); A[-1] := 1; A[0] := 1; for n to len-2 do A[n] := (2*n-1)*A[n-1]+add(A[j]*A[n-j-1], j=0..n-1) od: convert(A, list) end: a_list(20); # Peter Luschny, Jul 18 2017

a[n_] := a[n] = (2n - 1)!! - Sum[ a[n - k](2k - 1)!!, {k, n-1}]; Array[a, 18, 0] (* Ignacio D. Peixoto, Jun 23 2006 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ 2 - 1 / Sum[ (2 k - 1)!! x^k, {k, 0, n}], {x, 0, n}]]; (* Michael Somos, Nov 16 2011 *)
a[n_]:= SeriesCoefficient[1+x(1/x+(E^((1/2)/x) Sqrt[2/\[Pi]] Sqrt[-(1/x)])/Erfc[Sqrt[-(1/x)]/Sqrt[2]]), {x,0,n}, Assumptions -> x >0](* Robert Coquereaux, Sep 14 2014 *)
max = 20; g = t/Fold[1 - ((t + #2)*z)/#1 &, 1, Range[max, 1, -1]]; T[n_, k_] := SeriesCoefficient[g, {z, 0, n}, {t, 0, k}]; a[0] = 1; a[n_] := Sum[T[n-1, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2016, after Philippe Deléham *)

{a(n) = if( n<0, 0, polcoeff( 2 - 1 / sum( k=0, n, x^k * (2*k)! /(2^k * k!), x * O(x^n)), n))}; /* Michael Somos, Feb 08 2011 */

{a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

from sympy import factorial2, cacheit
@cacheit
def a(n): return 1 if n == 0 else factorial2(2*n - 1) - sum(factorial2(2*k - 1)*a(n - k) for k in range(1, n))
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 18 2017

G.f.: 2 - 1/(1 + Sum_{n>=1} (2*n-1)!! * x^n ).
a(n+1) = Sum_{k=0..n} A089949(n, k)*2^k. - Philippe Deléham, Aug 15 2005
a(n+1) = Sum_{k=0..n} A053979(n,k). - Philippe Deléham, Mar 24 2007
From Paul Barry, Nov 26 2009: (Start)
G.f.: 1+x/(1-2x/(1-3x/(1-4x/(1-5x/(1-6x/(1-... (continued fraction).
G.f.: 1+x/(1-2x-6x^2/(1-7x-20x^2/(1-11x-42x^2/(1-15x-72x^2/(1-19x-110x^2/(1-... (continued fraction). (End)
G.f.: 1 + x * B(x) * C(x) where B(x) is the g.f. for A001147 and C(x) is the g.f. for A005416. - Michael Somos, Feb 08 2011
G.f.: 1+x/W(0); where W(k)=1+x+x*2k-x*(2k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
From Peter Bala, Dec 22 2011: (Start)
Recurrence relation: a(n+1) = (2*n-1)*a(n) + Sum_{k = 1..n} a(k)*a(n+1-k) for n >= 0 and a(1) = 1.
The o.g.f. B(x) = Sum_{n>=1} a(n)*x^(2*n-1) = x + 2*x^3 + 10*x^5 + 74*x^7 + ... satisfies the Riccati differential equation y'(x) = -1/x^2 + (1/x^3)*y(x) - (1/x^2)*y(x)^2 with initial condition y(0) = 0 (cf. A005412). The solution is B(x) = 1/z(x) + 1/x, where z(x) = -Sum_{n>=0} A001147(n) * x^(2*n+1) = -(x + x^3 + 3*x^5 + 15*x^7 + ...). The function b(x) = -B(1/x) satisfies b'(x) = -1 - (x + b(x))*b(x). Hence the differential operator (D^2 + x*D + 1), where D = d/dx, factorizes as (D - a(x))*(D - b(x)), where a(x) = -(x + b(x)), as conjectured by [Edgar, Problem 4.32]. For a refinement of this sequence see A053979. (End)
From Sergei N. Gladkovskii, Aug 19 2012, Oct 24 2012, Mar 19 2013, May 20 2013, May 29 2013, Aug 04 2013, Aug 05 2013: (Start)
Continued fractions:
G.f.: 2 - G(0) where G(k) = 1 - (k+1)*x/G(k+1).
G.f.: 2 - U(0) where U(k) = 1 - (2*k+1)*x/(1 - (2*k+2)*x/U(k+1)).
G.f.: 2 - U(0) where U(k) = 1 - (4*k+1)*x - (2*k+1)*(2*k+2)*x^2/U(k+1).
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+2)/(1 - x*(2*k+3)/Q(k+1)).
G.f.: 1 + x/Q(0) where Q(k) = 1 - x*(k+2)/Q(k+1).
G.f.: 2 - G(0)/2 where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+2)/ G(k+1))).
G.f.: 1 + x*G(0) where G(k) = 1 - x*(k+2)/(x*(k+2) - 1/G(k+1)).
G.f.: 2 - 1/B(x) where B(x) is the g.f. of A001147.
G.f.: 1 + x/(1-2*x*B(x)) where B(x) is the g.f. of A167872. (End)
a(n) ~ 2^(n+1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 10 2014
G.f.: 1 + x*(1/x + (sqrt(2/Pi) * exp(1/(2*x)) * sqrt(-1/x))/Erfc(sqrt(-1/x)/sqrt(2))) where Erfc(z) = 1 - Erf(z) is the complementary error function, and Erf(z) is the integral of the Gaussian distribution. This generating function is obtained from the generating functional of (4-dimensional) QED, evaluated in dimension 0 for the 2-point function, without the modification implementing Furry theorem. - Robert Coquereaux, Sep 14 2014
From Peter Bala, May 23 2017: (Start)
G.f. A(x) = 1 + x/(1 + x - 3*x/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - ...)))).
A(x) = 1 + x/(1 + x - 3*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - ...))))))). (End)

Formula corrected by Ignacio D. Peixoto, Jun 23 2006

More terms from Sean A. Irvine, Feb 27 2011

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

Original entry on oeis.org

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855

Offset: 0

N. J. A. Sloane

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
Number of uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
Handbook of Combinatorics, North-Holland '95, p. 891.
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).
W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

T. D. Noe, Table of n, a(n) for n = 0..200
E. A. Bender and N. C. Wormald, The number of loopless planar maps, Discr. Math. 54:2 (1985), 235-237.
Andreas Bernig, Unitarily invariant valuations and Tutte's sequence, arXiv:2001.03372 [math.DG], 2020.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \overline{T}$, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. (2020) Vol. 8, 052.
Enrica Duchi and Corentin Henriet, A bijection between Tamari intervals and extended fighting fish, arXiv:2206.04375 [math.CO], 2022.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Grégory Chatel and Viviane Pons, Counting smaller elements in the Tamari and m-Tamari lattices, arXiv preprint arXiv:1311.3922 [math.CO], 2013.
François David, A model of random surfaces with nontrivial critical behavior, Nuclear Physics B, v. 257 (1985), 543-576.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Wenjie Fang, Planar triangulations, bridgeless planar maps and Tamari intervals, arXiv:1611.07922 [math.CO], 2016.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
C. Germain and J. Pallo, The number of coverings in four Catalan lattices, Intern. J. Computer Math., Vol. 61 (1996) pp. 19-28. (See p. 27.)
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Viviane Pons, Combinatorics of the Permutahedra, Associahedra, and Friends, arXiv:2310.12687 [math.CO], 2023. See p. 37.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. See Eq. 5.12.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105-126. MR0572468 (81j:05073).
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq. (6).
Noam Zeilberger, A sequent calculus for the Tamari order, arXiv:1701.02917 [cs.LO], 2017.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. See also Part 2.

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

[Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016

A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:

Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)

{a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */

{a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */

def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12,  2/12], [ 3/12,  9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12,  5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Edited by F. Chapoton, Feb 03 2011

The limit of (a(n+1)/a(n))/n = 2. - Paul D. Hanna, Aug 08 2014

The g.f. of related sequence A000699, where A000699(n) is the number of irreducible diagrams with 2n nodes, satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].

Partial sums of number of irreducible diagrams (in the sense of perturbation expansion in quantum field theory: spinor case in 4 spacetime dimensions) with 2n nodes. The subsequence of primes in this partial sum begins: 2, 281, and there never seem to be any more, as the underlying sequence is all even after A000699(8).

cp's OEIS Frontend

A280777 Denominators of coefficients in asymptotic expansion of C_n (number of connected chord diagrams, A000699).

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A178685 G.f.: A(x) = x/Series_Reversion(G(x)) where x*A(G(x)) = G(x) is the g.f. of A000699.

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A280776 Numerators of coefficients in asymptotic expansion of C_n (number of connected chord diagrams, A000699).

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A212273 a(n) = n * A000699(n).

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A304685 a(n) = A000699(n) (mod 3).

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A051863 Duplicate of A000699.

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A174432 Partial sums of A000699.

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A036987 Fredholm-Rueppel sequence.

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