A053979 -id:A053979 - cp's OEIS frontend

A305873 Coefficients of polynomials g_b(x) that arise in the generating function for rooted maps (A053979).

1, 3, 5, 15, 65, 60, 105, 804, 1730, 1105, 945, 10824, 39110, 55645, 27120, 10395, 162357, 854250, 1987270, 2105070, 828250, 135135, 2714445, 19180410, 63897550, 108878610, 91692550, 30220800, 2027025, 50301360, 452984532, 2004435096, 4836052370, 6479714440, 4523710100, 1282031525

Offset: 1

R. J. Mathar, Jun 12 2018

The generating function of the b-th subdiagonal of A053979 is g_b(y)*(1-sqrt(1-4x))/2/(1-4x)^b, b>=0, where g_b(y) = 1 (b= 0 or 1), 3+5*y (b=2), 15+65*y+60*y^2 (b=3) etc are the coefficients in this table, and where y=(1/sqrt(1-4x)-1)/2.

The coefficient 794 cited by Walsh-Lehman (1972) has been corrected to 804.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I, J. Comb. Theory B 13 (1972), 192-218, eq. (5a).

Cf. A062980 (diagonal), A001147 (first column)

A305873:= proc(b,x)
    local gn1,k ;
    option remember;
    if b = 0 or b= 1 then
        return 1 ;
    else
        gn1 := procname(b-1,x) ;
        add(procname(k,x)*procname(b-k,x),k=1..b-1) ;
        gbx := %*x+(2*(b-1)*(1+2*x)+1)*gn1 ;
        expand(gbx+2*x*(x+1)*diff(gn1,x)) ;
    end if;
end proc:
for b from 1 to 8 do
    gx := A305873(b,x) ;
    for l from 0 to b-1 do
        printf("%d,",coeff(gx,x,l)) ;
    end do:
    printf("\n") ;
end do:

A305873[b_, x_] := A305873[b, x] = Module[{gn1, k, s}, If[b == 0 || b == 1, Return@1, gn1 = A305873[b - 1, x]; s = Sum[A305873[k, x]*A305873[b - k, x], {k, 1, b - 1}]; gbx = s*x + (2*(b - 1)*(1 + 2*x) + 1)*gn1; Expand[gbx + 2*x*(x + 1)*D[gn1, x]]]];
Reap[For[b = 1, b <= 8, b++, gx = A305873[b, x]; For[l = 0, l <= b - 1, l++, Sow[Coefficient[gx, x, l]]]]][[2, 1]] (* Jean-François Alcover, Nov 09 2023, after Maple program *)

A000346 a(n) = 2^(2n+1) - binomial(2n+1, n+1).

Original entry on oeis.org

1, 5, 22, 93, 386, 1586, 6476, 26333, 106762, 431910, 1744436, 7036530, 28354132, 114159428, 459312152, 1846943453, 7423131482, 29822170718, 119766321572, 480832549478, 1929894318332, 7744043540348, 31067656725032, 124613686513778, 499744650202436

Offset: 0

N. J. A. Sloane, Wolfdieter Lang

Also a(n) = 2nd elementary symmetric function of binomial(n,0), binomial(n,1), ..., binomial(n,n).
Also a(n) = one half the sum of the heights, over all Dyck (n+2)-paths, of the vertices that are at even height and terminate an upstep. For example with n=1, these vertices are indicated by asterisks in the 5 Dyck 3-paths: UU*UDDD, UU*DU*DD, UDUU*DD, UDUDUD, UU*DDUD, yielding a(1)=(2+4+2+0+2)/2=5. - David Callan, Jul 14 2006
Hankel transform is (-1)^n*(2n+1); the Hankel transform of sum(k=0..n, C(2*n,k) ) - C(2n,n) is (-1)^n*n. - Paul Barry, Jan 21 2007
Row sums of the Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/2) (A187926). - Emanuele Munarini, Mar 16 2011
From Gus Wiseman, Jul 19 2021: (Start)
For n > 0, a(n-1) is also the number of integer compositions of 2n with nonzero alternating sum, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These compositions are ranked by A053754 /\ A345921. For example, the a(3-1) = 22 compositions of 6 are:
  (6)  (1,5)  (1,1,4)  (1,1,1,3)  (1,1,1,1,2)
       (2,4)  (1,2,3)  (1,1,3,1)  (1,1,2,1,1)
       (4,2)  (1,4,1)  (1,2,1,2)  (2,1,1,1,1)
       (5,1)  (2,1,3)  (1,3,1,1)
              (2,2,2)  (2,1,2,1)
              (3,1,2)  (3,1,1,1)
              (3,2,1)
              (4,1,1)
(End)

			G.f. = 1 + 5*x + 22*x^2 + 93*x^3 + 386*x^4 + 1586*x^5 + 6476*x^6 + ...

T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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..500
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
Vijay Balasubramanian, Javier M. Magan, and Qingyue Wu, A Tale of Two Hungarians: Tridiagonalizing Random Matrices, arXiv:2208.08452 [hep-th], 2022.
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
E. A. Bender, E. R. Canfield and R. W. Robinson, The enumeration of maps on the torus and the projective plane, Canad. Math. Bull., 31 (1988), 257-271; see p. 270.
D. E. Davenport, L. K. Pudwell, L. W. Shapiro and L. C. Woodson, The Boundary of Ordered Trees, 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 185
R. K. Guy, Letter to N. J. A. Sloane
Toufik Mansour and José L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7. - From _N. J. A. Sloane_, Nov 25 2012
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (A_n for s=2).
Vera Posch, Correlators in Matrix Models, Master Thesis, Uppsala Univ. (Sweden 2023). See p. 44.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
W. T. Tutte, On the enumeration of planar maps. Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218 (eq. 5, b=1).
N. J. A. Sloane, Notes

Cf. A000108, A014137, A014318. A column of A058893. Subdiagonal of A053979.
Bisection of A058622 and (possibly) A007008.
Cf. A045621, A068551.
Cf. A033504, A038806, A055217, A069731.
Even bisection of A294175 (without the first two terms).
The following relate to compositions of 2n with alternating sum k.
- The k > 0 case is counted by A000302.
- The k <= 0 case is counted by A000302.
- The k != 0 case is counted by A000346 (this sequence).
- The k = 0 case is counted by A001700 or A088218.
- The k < 0 case is counted by A008549.
- The k >= 0 case is counted by A114121.
A011782 counts compositions.
A086543 counts partitions with nonzero alternating sum (bisection: A182616).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A345197 counts compositions by length and alternating sum.
Cf. A000070, A001791, A007318, A025047, A027306, A032443, A053754, A120452, A163493, A239830, A344611, A345921.

[2^(2*n+1) - Binomial(2*n+1,n+1): n in [0..30]]; // Vincenzo Librandi, Jun 07 2011

seq(2^(2*n+1)-binomial(2*n,n)*(2*n+1)/(n+1), n=0..12); # Emanuele Munarini, Mar 16 2011

Table[2^(2n+1)-Binomial[2n,n](2n+1)/(n+1),{n,0,20}] (* Emanuele Munarini, Mar 16 2011 *)
a[ n_] := If[ n<-4, 0, (4^(n + 1) - Binomial[2 n + 2, n + 1]) / 2]; (* Michael Somos, Jan 25 2014 *)

makelist(2^(2*n+1)-binomial(2*n,n)*(2*n+1)/(n+1),n,0,12); /* Emanuele Munarini, Mar 16 2011 */

{a(n) = if( n<-4, 0, n++; (2^(2*n) - binomial(2*n, n)) / 2)}; /* Michael Somos, Jan 25 2014 */

G.f.: c(x)/(1-4x), c(x) = g.f. of Catalan numbers.
Convolution of Catalan numbers and powers of 4.
Also one half of convolution of central binomial coeffs. A000984(n), n=0, 1, 2, ... with shifted central binomial coeffs. A000984(n), n=1, 2, 3, ...
a(n) = A045621(2n+1) = (1/2)*A068551(n+1).
a(n) = Sum_{k=0..n} A000984(k)*A001700(n-k). - Philippe Deléham, Jan 22 2004
a(n) = Sum_{k=0..n+1} binomial(n+k, k-1)2^(n-k+1). - Paul Barry, Nov 13 2004
a(n) = Sum_{i=0..n} binomial(2n+2, i). See A008949. - Ed Catmur (ed(AT)catmur.co.uk), Dec 09 2006
a(n) = Sum_{k=0..n+1, C(2n+2,k)} - C(2n+2,n+1). - Paul Barry, Jan 21 2007
Logarithm g.f. log(1/(2-C(x)))=sum(n>0, a(n)/n*x^n), C(x)=(1-sqrt(1-4*x))/2x (A000108). - Vladimir Kruchinin, Aug 10 2010
D-finite with recurrence: (n+3) a(n+2) - 2(4n+9) a(n+1) + 8(2n+3) a(n) = 0. - Emanuele Munarini, Mar 16 2011
E.g.f.:exp(2*x)*(2*exp(2*x) - BesselI(0,2*x) - BesselI(1,2*x)).
This is the first derivative of exp(2*x)*(exp(2*x) - BesselI(0,2*x))/2. See the e.g.f. of A032443 (which has a plus sign) and the remarks given there. - Wolfdieter Lang, Jan 16 2012
a(n) = Sum_{0<=iMircea Merca, Apr 05 2012
A000346 = A004171 - A001700. See A032443 for the sum. - M. F. Hasler, Jan 02 2014
0 = a(n) * (256*a(n+1) - 224*a(n+2) + 40*a(n+3)) + a(n+1) * (-32*a(n+1) + 56*a(n+2) - 14*a(n+3)) + a(n+2) * (-2*a(n+2) + a(n+3)) if n>-5. - Michael Somos, Jan 25 2014
REVERT transform is [1,-5,28,-168,1056,...] = alternating signed version of A069731. - Michael Somos, Jan 25 2014
Convolution square is A038806. - Michael Somos, Jan 25 2014
BINOMIAL transform of A055217(n-1) is a(n-1). - Michael Somos, Jan 25 2014
(n+1) * a(n) = A033504(n). - Michael Somos, Jan 25 2014
Recurrence: (n+1)*a(n) = 512*(2*n-7)*a(n-5) + 256*(13-5*n)*a(n-4) + 64*(10*n-17)*a(n-3) + 32*(4-5*n)*a(n-2) + 2*(10*n+1)*a(n-1), n>=5. - Fung Lam, Mar 21 2014
Asymptotic approximation: a(n) ~ 2^(2n+1)*(1-1/sqrt(n*Pi)). - Fung Lam, Mar 21 2014
a(n) = Sum_{m = n+2..2*(n+1)} binomial(2*(n+1), m), n >= 0. - Wolfdieter Lang, May 22 2015
a(n) = A000302(n) + A008549(n). - Gus Wiseman, Jul 19 2021
a(n) = Sum_{j=1..n+1} Sum_{k=1..j} 2^(j-k)*binomial(n+k-1, n). - Fabio Visonà, May 04 2022
a(n) = (1/2)*(-1)^n*binomial(-(n+1), n+2)*hypergeom([1, 2*n + 3], [n + 3], 1/2). - Peter Luschny, Nov 29 2023

Corrected by Christian G. Bower

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.
A. Prunotto, W. M. Alberico, and P. Czerski, Feynman diagrams and rooted maps, Open Phys. 16 (2018) 149-167.
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.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003.

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

A380622 Array read by antidiagonals: T(n,k) is the number of rooted k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 3, 5, 1, 0, 1, 0, 24, 0, 1, 0, 1, 15, 189, 297, 60, 1, 0, 1, 0, 1695, 0, 4896, 0, 1, 0, 1, 105, 19305, 472200, 869400, 100278, 1105, 1, 0, 1, 0, 252000, 0, 242183775, 0, 2450304, 0, 1, 0, 1, 945, 3828825, 2465608950, 103694490900, 198147676875, 16482741030, 69533397, 27120, 1, 0

Offset: 0

Andrew Howroyd, Jan 29 2025

The combinatorial maps considered are connected, unlabeled, may have loops and parallel edges and are of any orientable genus.

			Array begins:
============================================================
n\k | 1 2    3       4      5         6     7          8 ...
----+-------------------------------------------------------
  0 | 1 1    1       1      1         1     1          1 ...
  1 | 0 1    0       3      0        15     0        105 ...
  2 | 1 1    5      24    189      1695 19305     252000 ...
  3 | 0 1    0     297      0    472200     0 2465608950 ...
  4 | 0 1   60    4896 869400 242183775 ...
  5 | 0 1    0  100278      0 ...
  6 | 0 1 1105 2450304 ...
  7 | 0 1    0 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns 2..6 are A000012, A062980 (with interspersed zeros), A292186, A380623 (with interspersed zeros), A380624.

Cf. A053979, A269919, A380625.

T(n,k)={my(A=O(x^(n*k+1)), g=serlaplace(serconvol(exp(x^k/k + A), exp(x^2/2 + A)))); polcoef(1 + x*deriv(g)/g, n*k)}

A380625(n) = Sum_{d|2*n} T(d,2*n/d).

A128709 O.g.f.: A(x) = 1/(1-1x/(1-3x/(1-5x/(1-7x/(1-...-(2n-1)*x/(1-...)))))) (continued fraction).

Original entry on oeis.org

1, 1, 4, 31, 364, 5746, 113944, 2719291, 75843724, 2420160286, 86941080904, 3471911602006, 152562875644984, 7315129181611876, 380045172886143664, 21266347877729314771, 1275148311699896290444, 81563275661324271278566

Offset: 0

Paul D. Hanna, Mar 23 2007

nonn

Hankel transform is A168440. - Paul Barry, Nov 25 2009

			G.f.: A(x) = 1 + x + 4x^2 + 31x^3 + 364x^4 + 5746x^5 + ...;
A(x) = 1/(1 - x*(1 + 3x + 24x^2 + 297x^3 + 4896x^4 + ...));
A(x) = 1/(1 - x/(1 - 3x*(1 + 5x + 60x^2 + 1035x^3 + 22500x^4 + ...)));
A(x) = 1/(1 - x/(1 - 3x/(1 - 5x*(1 + 7x + 112x^2 + 2485x^3 + ...)))).

nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[(2*Range[nmax + 1]-1)*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2017 *)

{a(n)=local(CF=1+x*O(x^n));for(k=0,n,CF=1/(1-(2*n-2*k+1)*x*CF));polcoeff(CF,n,x)}

a(n) = Sum_{k=0..n} (-1)^k*2^(n-k)*A053979(n,k). - Philippe Deléham, Mar 24 2007
a(n) = Sum_{k=0..n} A094344(n,k)*3^(n-k). - Philippe Deléham, Mar 27 2007
G.f.: 1/(1-x-3x^2/(1-8x-35x^2/(1-16x-99x^2/(1-24x-195x^2/(1-32x-323x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = top left term of M^n, n > 0; M = the infinite square production matrix:
  1, 3, 0, 0, ...
  1, 3, 5, 0, ...
  1, 3, 5, 7, ...
  ...
Also, a(n+1) = sum of top row terms of M^n. Example: top row of M^3 = (31, 93, 135, 105, 0, 0, 0, ...), where a(3) = 31 and a(4) = 364 = (31 + 93 + 135 + 105). - Gary W. Adamson, Jul 14 2011
G.f.: 1/T(0) where T(k) = 1 - x*(4*k+1)/(1 - x*(4*k+3)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: G(0), where G(k) = 1 - x*(2*k+1)/(x*(2*k+1) - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 05 2013
a(n) ~ 2^(2*n - 1/2) * (n-1)! / Pi. - Vaclav Kotesovec, Aug 24 2017

A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 18, 38, 34, 14, 3, 105, 275, 288, 154, 42, 6, 902, 2614, 3102, 1959, 705, 140, 14, 9749, 30346, 39242, 27898, 11956, 3142, 473, 34, 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95, 1915951, 6513999, 9484003, 7911844, 4230802, 1523176, 373188, 60614, 5969, 280

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of sensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |      1       2       3       4       5      6      7     8   9
----+----------------------------------------------------------------
  0 |      1
  1 |      1,      1
  2 |      2,      2,      1;
  3 |      5,      8,      5,      2;
  4 |     18,     38,     34,     14,      3;
  5 |    105,    275,    288,    154,     42,     6;
  6 |    902,   2614,   3102,   1959,    705,   140,    14;
  7 |   9749,  30346,  39242,  27898,  11956,  3142,   473,   34;
  8 | 127072, 415360, 573654, 446078, 217000, 68544, 13886, 1670, 95;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A170946.
Main diagonal is A002995(n+1).
Second diagonal gives A380237.
Columns 1..3 are A007769, A380618, A380619.
Cf. A053979 (rooted), A379430 (planar), A380616 (unsensed), A380617 (achiral).

InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
b(k,r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
C(k,r,y)={my(p=sumdiv(k,d,eulerphi(k/d)*y^d)/k); sum(i=0, r, abs(stirling(r,i,1))*p^i)/r!}
S(n,k,y)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k,r)*C(k,r,y)), O(x*x^n))}
G(n,y='y)={prod(k=1, 2*n, S(n,k,y))}
T(n)={[Vecrev(p/y) | p<-Vec(y+InvEulerMTS(G(n)))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

A380616 Triangle read by rows: T(n,k) is the number of unsensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 8, 5, 2, 17, 33, 30, 13, 3, 79, 198, 208, 118, 35, 6, 554, 1571, 1894, 1232, 472, 104, 12, 5283, 16431, 21440, 15545, 6879, 1914, 315, 27, 65346, 213831, 296952, 233027, 115134, 37311, 7881, 1021, 65, 966156, 3288821, 4799336, 4019360, 2163112, 787065, 196267, 32857, 3407, 175

Offset: 0

Andrew Howroyd, Jan 28 2025

By duality, also the number of unsensed combinatorial maps with n edges and k faces.

			Triangle begins:
n\k |     1       2       3       4       5      6     7     8   9
----+--------------------------------------------------------------
  0 |     1;
  1 |     1,      1;
  2 |     2,      2,      1;
  3 |     5,      8,      5,      2;
  4 |    17,     33,     30,     13,      3;
  5 |    79,    198,    208,    118,     35,     6;
  6 |   554,   1571,   1894,   1232,    472,   104,   12;
  7 |  5283,  16431,  21440,  15545,   6879,  1914,  315,   27;
  8 | 65346, 213831, 296952, 233027, 115134, 37311, 7881, 1021, 65;
  ...

Row sums are A214816.
Main diagonal is A006082(n+1).
Columns 1..3 are A054499, A380620, A380621.
Cf. A053979 (rooted), A277741 (planar), A380615 (sensed), A380617 (achiral).

T(n,k) = (A380615(n,k) + A380617(n,k))/2.

A127160 Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,2,3,4,5,6,...] DELTA [1,1,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 3, 7, 5, 0, 15, 39, 37, 14, 0, 105, 296, 326, 176, 42, 0, 945, 2838, 3458, 2228, 794, 132, 0, 10395, 32859, 43191, 31235, 13553, 3473, 429, 0, 135135, 445767, 622259, 489899, 241225, 76417, 14893, 1430

Offset: 0

Philippe Deléham, Mar 25 2007

This triangle enumerates fixed-point-free involutions in S_n by number of left-to-right maxima. For instance there are 15 fixed point free involutions on 6 elements: 3 have 1 left to right maxima, namely (1,6)(2,3)(4,5), (1,6)(2,4)(3,5) and (3,6)(2,5)(3,4); 7 have 2 left-to right maxima and 5 have 3 left to right maxima. - Robert Cori (rcori(AT)cs.brown.edu), Apr 25 2008

A053979*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 06 2012

			Triangle begins:
1;
0, 1;
0, 1, 2;
0, 3, 7, 5;
0, 15, 39, 37, 14;
0, 105, 296, 326, 176, 42;
0, 945, 2838, 3458, 2228, 794, 132;
0, 10395, 32859, 43191, 31235, 13553, 3473, 429;
0, 135135, 445767, 622259, 489899, 241225, 76417, 14893, 1430;

A053979.

nmax = 8;
DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x r[[k + 1]] + y s[[k + 1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k] p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[ p[n, 0], x^(n - k) y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
DELTA[Range[0, nmax], Table[1, {nmax+1}], nmax] // Flatten (* Jean-François Alcover, Nov 12 2019 *)

Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000108(n), A001147(n), A128709(n) for x = -1,0,1,2 respectively.
From Peter Bala, Dec 22 2011: (Start)
The o.g.f. in the form G(x,t) = x/(1 - t*x^2/(1 - (t+1)*x^2/(1 - (t+2)*x^2/(1 - (t+3)*x^2/(1 - ... ))))) = x + t*x^3 + (t+2*t^2)*x^5 + ... satisfies the Riccati differential equation (1 - (t-1)*x*G)*G = x + x^3*dG/dx. Writing G(x,t) = sum {n = 1..inf} R(n,t)*x^(2*n-1), the row generating polynomials R(n,t) satisfy the recurrence R(n+1,t) = (2*n-1)*R(n,t) + (t-1)*sum {k = 1..n} R(k,t)*R(n+1-k,t) with initial condition R(1,t) = 1.
G(x,t+1) = x + (1+t)*x^3 + (3+5*t+2*t^2)*x^5 + ... is an o.g.f. for A053979.
(End)

Corrected and extended by Peter Bala, Dec 20 2011

cp's OEIS Frontend

A305873 Coefficients of polynomials g_b(x) that arise in the generating function for rooted maps (A053979).

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A000346 a(n) = 2^(2*n+1) - binomial(2*n+1, n+1).

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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!.

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A380622 Array read by antidiagonals: T(n,k) is the number of rooted k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

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A128709 O.g.f.: A(x) = 1/(1-1*x/(1-3*x/(1-5*x/(1-7*x/(1-...-(2n-1)*x/(1-...)))))) (continued fraction).

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A380615 Triangle read by rows: T(n,k) is the number of sensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

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A380616 Triangle read by rows: T(n,k) is the number of unsensed combinatorial maps with n edges and k vertices, 1 <= k <= n + 1.

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A000346 a(n) = 2^(2n+1) - binomial(2n+1, n+1).

A128709 O.g.f.: A(x) = 1/(1-1x/(1-3x/(1-5x/(1-7x/(1-...-(2n-1)*x/(1-...)))))) (continued fraction).