A005568 -id:A005568 - cp's OEIS frontend

A168452 Self-convolution of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).

1, 4, 24, 180, 1556, 14840, 152092, 1646652, 18613664, 217852008, 2623657384, 32361812912, 407342311632, 5217211974832, 67836910362772, 893766246630572, 11913422912188432, 160450066324972472, 2181014117345997704, 29894260817385950064, 412839378639052110464

Offset: 0

Paul D. Hanna, Nov 26 2009

nonn

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 180*x^3 + 1556*x^4 + 14840*x^5 +...
A(x)^(1/2) = 1 + 2*x + 10*x^2 + 70*x^3 + 588*x^4 + 5544*x^5 + 56628*x^6 +...+ A005568(n)*x^n +...
A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A004304:
G(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...+ A004304(n)*x^n +...
G(x)^2 = 1 + 4*x + 8*x^2 + 20*x^3 + 84*x^4 + 456*x^5 + 2860*x^6 +...+ A168451(n)*x^n +...

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A168451, A004304, A005568, A000108, variant: A168358.

a:= proc(n) option remember; `if`(n<3, [1, 4, 24][n+1],
      (12*n*(n+1)*(16*n^4+68*n^3+44*n^2-63*n-25) *a(n-1)
       -(3072*n^6+768*n^5-8448*n^4+1152*n^3+3264*n^2-288) *a(n-2)
       +1024*n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(4*n+1) *a(n-3)) /
      ((n+1)^2*(n+2)*(n+3)*(n+4)*(4*n-3)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 20 2013

c[n_] := CatalanNumber[n]*CatalanNumber[n+1]; a[n_] := ListConvolve[cc = Array[c, n+1, 0], cc][[1]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 22 2017 *)

{a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)*(binomial(2*m,m)/(m+1))));polcoeff(Ser(C_2)^2,n)}

G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)^2) where G(x) = g.f. of A004304, where A004304(n) is the number of planar tree-rooted maps with n edges.
G.f.: A(x) = G(x*A(x))^2 where A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A004304.
a(n) ~ c * 16^n / n^3, where c = 3.07968404... . - Vaclav Kotesovec, Sep 12 2014
Conjecture D-finite with recurrence 3*(n+4)*(n+3)*(n+2)*(n+1)^2*a(n) -4*n*(n+1) *(32*n^3+164*n^2+233*n+75)*a(n-1) +96*(16*n^5+24*n^4-14*n^3-28*n^2-16*n+3) *a(n-2) +1536*(-8*n^4+22*n^3-32*n+15)*a(n-3) -16384*(2*n-5)*(n-1)*(n-2)*(n-3)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Nov 22 2024

Typo in formula corrected by Paul D. Hanna, Nov 28 2009

A005817 a(n) = C(floor(n/2 + 1/2))*C(floor(n/2 + 1)) where C(i) = Catalan numbers A000108.

Original entry on oeis.org

1, 1, 2, 4, 10, 25, 70, 196, 588, 1764, 5544, 17424, 56628, 184041, 613470, 2044900, 6952660, 23639044, 81662152, 282105616, 987369656, 3455793796, 12228193432, 43268992144, 154532114800, 551900410000, 1986841476000, 7152629313600

Offset: 0

Simon Plouffe and N. J. A. Sloane

Number of lattice paths in the first quadrant that do not cross the main diagonal, go from (0,0) to a point on the x-axis and consist of n+1 steps from the set {E=(1,0), W=(-1,0), N=(0,1), S=(0,-1)}. Example: a(2)=4 because we have EEE, ENS, EEW and EWE [Gouyou-Beauchamps]. - Emeric Deutsch, Apr 29 2004
Also, number of standard Young tableaux of height <= 4. - Mike Zabrocki, Mar 24 2007
Also, number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, 1), (0, -1), (0, 1), (1, -1)}. - Manuel Kauers, Nov 18 2008
Also, number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (0, -1, 1), (0, 1, 0), (1, 0, -1)} - Manuel Kauers, Nov 18 2008
Also, number of n-length words w over alphabet {a,b,c,d} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c)>= #(z,d), where #(z,x) counts the letters x in word z.  The a(4) = 10 words are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca, abcd. - Alois P. Heinz, May 30 2012
Also, for n>0, number of coalescent histories for a maximally symmetric matching bicaterpillar gene tree and species tree with n+1 leaves, that is, a bicaterpillar divided into caterpillars of size floor(n/2+1/2) and floor(n/2+1) leaves (Rosenberg 2007, Theorem 3.10). - Noah A Rosenberg, Feb 04 2019

			There are 26 standard tableaux of size 5, one of them is of length longer than 4 so a(5) = 25.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(b), y_4(n), p. 452.

T. D. Noe, Table of n, a(n) for n=0..200
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
R. Cori et al., Shuffle of parenthesis systems and Baxter permutations, J. Combin. Theory, A 43 (1986), 1-22.
D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.
Zhicong Lin and Jing Liu, Proof of Dilks' bijectivity conjecture on Baxter permutations, arXiv:2112.11698 [math.CO], 2021.
Alon Regev, Amitai Regev, and Doron Zeilberger, Identities in character tables of S_n, arXiv preprint arXiv:1507.03499 [math.CO], 2015.
N. A. Rosenberg, Counting coalescent histories, J. Comput. Biol. 14 (2007), 360-377.
Albert Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949. [Annotated scanned copy]

Cf. A000108, A001405, A001006, A005700, A049401, A007579, A007578.
Bisections are A001246 and A005568.
Column k=4 of A182172. - Alois P. Heinz, May 30 2012

[Catalan(n div 2)*Catalan(((n+1)) div 2) : n in [1..30]]; // Vincenzo Librandi, Apr 16 2019

c := n->binomial(2*n,n)/(n+1); seq(c(floor((n+1)/2))*c(floor(n/2+1)), n=0..16);

Table[Binomial[2*Floor[(n+1)/2], Floor[(n+1)/2]]/(Floor[(n+1)/2]+1) * Binomial[2*Floor[n/2+1], Floor[n/2+1]]/(Floor[n/2+1]+1), {n,0,20}] (* Vaclav Kotesovec, Sep 11 2013 *)

c(n)=binomial(2*n, n)/(n+1)
for(n=1, 40, print1(c(floor((n+1)/2))*c(floor(n/2+1)), ", ")); \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

G.f.: (hypergeom([-1/2, -1/2],[1],16*x^2)-2*x*hypergeom([-1/2, 1/2],[2],16*x^2)-1+2*x-4*x^2)/(4*x^3). - Mark van Hoeij, Oct 25 2011
D-finite with recurrence (n+3)*(n+4)*a(n) = 4*(2*n+3)*a(n-1) + 16*(n-1)*n*a(n-2). - Vaclav Kotesovec, Sep 11 2013
a(n) ~ 2^(2*n+5)/(Pi*n^3). - Vaclav Kotesovec, Sep 11 2013

Description corrected Feb 15 1997.
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008
Offset changed by N. J. A. Sloane, Nov 28 2008

A064037 Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant.

Original entry on oeis.org

1, 3, 24, 285, 4242, 73206, 1403028, 29082339, 640672890, 14818136190, 356665411440, 8874875097270, 227135946200940, 5955171596514900, 159439898653636320, 4347741997166750235, 120493374240909299130, 3387806231071627372590, 96488484001399878973200

Offset: 0

Henry Bottomley, Aug 23 2001

nonn

			a(1)=3 and a(2)=24 since if the possible steps are Right, Left, Up, Down, Forwards and Backwards, then the two-step paths are FB, RL and UD, while the four-step paths are FBFB, FBRL, FBUD, FFBB, FRBL, FRLB, FUBD, FUDB, RFBL, RFLB, RLFB, RLRL, RLUD, RRLL, RUDL, RULD, UDFB, UDRL, UDUD, UFBD, UFDB, URDL, URLD, UUDD.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
James Mallos, A 6-Letter 'DNA' for Baskets with Handles, Mathematics (2019) Vol. 7, No. 2, 165.
G. Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211.

Cf. A064036. The two- and one-dimensional equivalents are A005568 and A000108.

f := -3*x+(1+sqrt(1-40*x+144*x^2))/4;
H := (1-2*f)*f*hypergeom([1/6,1/3],[1],27*(1-2*f)*f^2)^2/sqrt(1+6*f);
r2 := (1-4*x)*(36*x-1)*(1920*x^2+166*x+1)*x^2;
r1 := -(138240*x^4+7776*x^3+200*x^2-92*x-1)*x;
r0 := 19800*x^3+764*x^2-86*x-1;
ogf := (r2*diff(H,x,x)+r1*diff(H,x)+r0*H)/(5760*x^4) + 1/(2*x);
series(ogf, x=0, 30); # Mark van Hoeij, Apr 19 2013
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 2*n+1, ((8*n-4)*(5*n^2+10*n+3)
       *a(n-1)-36*(2*n-1)*(2*n-3)*(n-1)*a(n-2))/((n+1)*(n+2)*(n+3)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 29 2019

Table[Sum[Binomial[2*n, 2*j] * CatalanNumber[j] * CatalanNumber[j+1] * CatalanNumber[n-j], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 09 2019 *)

C(n,k) = binomial(n,k);
c(n) = binomial(2*n,n)/(n+1);
a(n) = sum(j=0,n, C(2*n, 2*j)*c(j)*c(j+1)*c(n-j));
/* Joerg Arndt, Apr 19 2013 */

a(n) = Sum_{j=0..n} C(2n, 2j)*c(j)*c(j+1)*c(n-j) where c(k)=A000108(k).
G.f. is a large expression in terms of hypergeometric functions and sqrt's, see Maple program. - Mark van Hoeij, Apr 19 2013
a(n) = binomial(2*n,n)*((7*n+11)*A002893(n+1)-(9*n+9)*A002893(n))/(2*(n+1)*(n+2)^2*(n+3)). - Mark van Hoeij, Apr 19 2013
a(n) ~ 2^(2*n - 2) * 3^(2*n + 9/2) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Jun 09 2019
D-finite with recurrence: (n+3)*(n+2)*(n+1)*a(n) -4*(2*n-1)*(5*n^2+10*n+3)*a(n-1) +36*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 20 2020

Added more terms, Joerg Arndt, Apr 19 2013

A000356 Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).

Original entry on oeis.org

1, 5, 35, 294, 2772, 28314, 306735, 3476330, 40831076, 493684828, 6114096716, 77266057400, 993420738000, 12964140630900, 171393565105575, 2291968851019650, 30961684478686500, 422056646314726500

Offset: 1

N. J. A. Sloane, Simon Plouffe

a(2n-1) is also the sum of the numbers of standard Young tableaux of size 2n+1 and of shapes (k+3,k+2,2^{n-2-k}), 0 <= k <= n-2. - Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010

Amitai Regev, Preprint. [From Amitai Regev (amitai.regev(AT)weizmann.ac.il), Mar 10 2010]
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 = 1..800
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Anatol N. Kirillov, Notes on Schubert, Grothendieck and key polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 034, 56 p. (2016).
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.

Cf. A000264, A000309.
Equals A005568/2.
Fourth row of array A102539.
Column of array A073165.
Image of A001700 under the "little Hankel" transform (see A056220 for definition). - John W. Layman, Aug 22 2000
Cf. A000891.

A000356 := proc(n)
    binomial(2*n,n)*binomial(2*n+1,n+1)/(n+1)/(n+2) ;
end proc:

CoefficientList[ Series[1 + (HypergeometricPFQ[{1, 3/2, 5/2}, {3, 4}, 16 x] - 1), {x, 0, 17}], x]
Table[(2*n)!*(2*n+2)!/(2*n!*(n+1)!^2*(n+2)!),{n,30}] (* Vincenzo Librandi, Mar 25 2012 *)

G.f.: (with offset 0) 3F2( [1, 3/2, 5/2], [3, 4], 16*x) = (1 - 2*x - 2F1( [-1/2, 1/2], [2], 16*x) ) / (4*x^2). - Olivier Gérard, Feb 16 2011
a(n)*(n+2) = A000891(n). - Gary W. Adamson, Apr 08 2011
D-finite with recurrence (n+2)*(n+1)*a(n)-4*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 03 2013
From Ilya Gutkovskiy, Feb 01 2017: (Start)
E.g.f.: (1/2)*(2F2(1/2,3/2; 2,3; 16*x) - 1).
a(n) ~ 2^(4*n+1)/(Pi*n^3). (End)
From Peter Bala, Feb 22 2023: (Start)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j - 1).
a(n) = (2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 3)/(i + j) for n >= 1.
 Cf. A003645. (End)

Better definition from Michael Albert, Oct 24 2008

A004304 Number of nonseparable planar tree-rooted maps with n edges.

Original entry on oeis.org

1, 2, 2, 6, 28, 160, 1036, 7294, 54548, 426960, 3463304, 28910816, 247104976, 2154192248, 19097610480, 171769942086, 1564484503044, 14407366963440, 133978878618904, 1256799271555872, 11881860129979440

Offset: 0

N. J. A. Sloane

nonn

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

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Dov Tamari, Monoïdes préordonnés et chaînes de Malcev, Bulletin de la Société Mathématique de France, Volume 82 (1954), 53-96. See end of Appendix II.
T. R. S. Walsh, A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259. See Table IVc.

Cf. A000264.

Cf. A005568, A168450, A168451, A168452. - Paul D. Hanna, Nov 26 2009

A004304 := proc(n) local N,x,ode ; Order := n+1 ; ode := x^2*diff(N(x),x,x)*(N(x)^3-16*x*N(x)) ; ode := ode + (x*diff(N(x),x))^3*(16-6*N(x)) ; ode := ode + (x*diff(N(x),x))^2*(12*N(x)^2-16*x-24*N(x)) ; ode := ode + x*diff(N(x),x)*(-8*N(x)^3+24*x*N(x)+12*N(x)^2) ; ode := ode + 2*N(x)^2*(N(x)^2-N(x)-6*x) ; dsolve({ode=0,N(0)=1,D(N)(0)=2},N(x),type=series) ; convert(%,polynom) ; rhs(%) ; RETURN( coeftayl(%,x=0,n)) ; end; for n from 0 to 20 do printf("%d,",A004304(n)) ; od ; # R. J. Mathar, Aug 18 2006

m = 22;
F[x_] = Sum[2 (2n+1) Binomial[2n, n]^2 x^n/((n+2)(n+1)^2), {n, 0, m}];
A[x_] = (x/InverseSeries[x F[x]^2 + O[x]^m, x])^(1/2);
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 28 2020 *)

{a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)*(binomial(2*m,m)/(m+1))));polcoeff((x/serreverse(x*Ser(C_2)^2))^(1/2),n)} \\ Paul D. Hanna, Nov 26 2009

seq(N) = {
  my(c(n)=binomial(2*n,n)/(n+1), s=Ser(apply(n->c(n)*c(n+1), [0..N])));
  Vec(subst(s, 'x, serreverse('x*s^2)));
};
seq(20)
\\ test: y=Ser(seq(200)); 0 == x^2*y''*(y^3 - 16*x*y) + (x*y')^3*(16-6*y) + (x*y')^2*(12*y^2-16*x-24*y) + x*y'*(-8*y^3 + 24*x*y + 12*y^2) + 2*y^2*(y^2-y-6*x)
\\ Gheorghe Coserea, Jun 13 2018

From Paul D. Hanna, Nov 26 2009: (Start)
G.f.: A(x) = [x/Series_Reversion(x*F(x)^2)]^(1/2) where F(x) = g.f. of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).
G.f.: A(x) = F(x/A(x)^2) where A(x*F(x)^2) = F(x) where F(x) = g.f. of A005568.
G.f.: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) where F(x) = g.f. of A168450.
G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = g.f. of A168450.
Self-convolution yields A168451.
(End)

More terms from R. J. Mathar, Aug 18 2006

A067640 Table T(n,k) giving number of two-legged knot diagrams with n >= 0 self-intersections and k >= 0 tangencies, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 8, 20, 10, 42, 174, 210, 70, 260, 1504, 2992, 2352, 588, 1796, 13300, 37100, 47820, 27720, 5544, 13396, 120744, 433620, 784672, 742296, 339768, 56628, 105706, 1122198, 4928798, 11515714, 15294006, 11376554, 4294290

Offset: 0

N. J. A. Sloane, Feb 05 2002

			Table begins
     1      2      10        70        588         5544         56628 ...
     2     20     210      2352      27720       339768       4294290 ...
     8    174    2992     47820     742296     11376554     173401952 ...
    42   1504   37100    784672   15294006    283730240    5095814988 ...
   260  13300  433620  11515714  271846056   5947557516  123429078160 ...
  1796 120744 4928798 158295072 4403552940 111289501120 2626033507768 ...
  ...

J. L. Jacobsen and P. Zinn-Justin, A Transfer Matrix approach to the Enumeration of Knots, arXiv:math-ph/0102015, 2001-2002.

Columns give A054993, A067641, A067642, A067643, rows give A005568, A067636, A067637, A067638, A067639.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 08 2002

A172392 a(n) = C(2n,n)*C(2n+2,n+1)/(n+2).

Original entry on oeis.org

1, 4, 30, 280, 2940, 33264, 396396, 4907760, 62573940, 816621520, 10861066216, 146738321184, 2008917492400, 27815780664000, 388924218927000, 5484594083378400, 77926940934668100, 1114620641232714000

Offset: 0

Paul D. Hanna, Feb 05 2010

nonn

			G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2940*x^4 + 33264*x^5 +...
A(x) = 1 + 2*2*x + 5*6*x^2 + 14*20*x^3 + 42*70*x^4 + 132*252*x^5 +...
where A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391:
A172391=[1,8,12,0,28,0,264,0,3720,0,63840,0,1232432,0,25731216,0,...].

Cf. A172391, A172393, A005568, A185248.

A172392 := n -> 4^n*coeff(simplify(hypergeom([3/2, -2*n], [3], -x)),x,n):
seq(A172392(n), n=0..17); # Peter Luschny, Feb 03 2015

CoefficientList[
Series[HypergeometricPFQ[{1/2, 3/2}, {3}, 16 x], {x, 0, 20}], x] (* From Olivier Gérard, Feb 15 2011 *)
Table[(Binomial[2n,n]Binomial[2n+2,n+1])/(n+2),{n,0,30}] (* Harvey P. Dale, Jul 16 2012 *)

{a(n)=binomial(2*n,n)*binomial(2*n+2,n+1)/(n+2)}

G.f. A(X) satisfies: A(x)^2 = G(x*A(x)^2) and G(x) = A(x/G(x))^2 = g.f. of A172391.
G.f. A(X) satisfies: A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) = g.f. of A172393.
a(n) = (n+1)*A005568(n) = A000108(n+1)*A000984(n), where A000108 is the Catalan numbers and A000984 is the central binomial coefficients.
G.f. : 2F1( (1/2, 3/2); (3))(16 x). - Olivier Gérard Feb 15 2011
a(n) = 4^n*[x^n]hypergeom([3/2, -2*n], [3], -x). - Peter Luschny, Feb 03 2015
D-finite with recurrence a(n) = a(n-1)*( 4*(4*n^2-1)/(n*(n+2)) ) for n>=1. - Peter Luschny, Feb 04 2015

A060897 Number of walks of length n on square lattice, starting at origin, staying in first and third quadrants.

Original entry on oeis.org

1, 4, 12, 44, 144, 528, 1808, 6676, 23536, 87568, 315136, 1180680, 4314560, 16263896, 60138816, 227899484, 850600944, 3238194560, 12177384544, 46542879384, 176110444736, 675431779856, 2568878867200, 9882068082112, 37747540858240, 145593279888736, 558190182662144

Offset: 0

David W. Wilson, May 05 2001

nonn

Is there a formula analogous to the (conjectured) formula for A060900?

Could be broken into the number of walks that are constrained to a quadrant and the number that cross the origin. (I.e., 2*A005566(n) + 2*A005566(n-2)*A005568(1) + 2*A005566(n-4)*A005568(2) + ... + All terms that cross the origin twice + three times + ... + Cross floor(n/2) times.) - Benjamin Phillabaum, Mar 13 2011

Alois P. Heinz, Table of n, a(n) for n = 0..750 (first 251 terms from Sean A. Irvine)

Cf. A005566, A005568, A001700, A060898, A060899, A060900.

\\ here B is A005566 and C is aerated A005568 as g.f.'s.
B(n)={sum(n=0, n, x^n*binomial(n, n\2)*binomial(n+1, (n+1)\2), O(x*x^n))}
C(n)={sum(n=0, (n+1)\2, x^(2*n)*binomial(2*n,n)*binomial(2*n+2,n+1)/((n+1)*(n+2)), O(x*x^n))}
seq(n) = {Vec( 1 + 2*(B(n)-1)/(2-C(n)) )} \\ Andrew Howroyd, Jan 05 2023

G.f.: 1 + 2*(B(x)-1)/(2 - C(x^2)) where B(x) is the g.f. of A005566 and C(x) is the g.f. of A005568. - Andrew Howroyd, Jan 05 2023

A168451 Self-convolution of A004304, where A004304(n) is the number of planar tree-rooted maps with n edges.

Original entry on oeis.org

1, 4, 8, 20, 84, 456, 2860, 19708, 145120, 1122680, 9023784, 74777248, 635292016, 5510485600, 48644137764, 435920025116, 3957758805776, 36345636909032, 337159090063880, 3155827384249824, 29776934546342464, 283001546964599248

Offset: 0

Paul D. Hanna, Nov 26 2009

nonn

			G.f.: A(x) = 1 + 4*x + 8*x^2 + 20*x^3 + 84*x^4 + 456*x^5 + 2860*x^6 +...
A(x)^(1/2) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...+ A004304(n)*x^n +...
G.f. satisfies: A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A005568:
F(x) = 1 + 2*x + 10*x^2 + 70*x^3 + 588*x^4 + 5544*x^5 + 56628*x^6 +...+ A000108(n)*A000108(n+1)*x^n +...
F(x)^2 = 1 + 4*x + 24*x^2 + 180*x^3 + 1556*x^4 + 14840*x^5 + 152092*x^6 +...+ A168452(n)*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..840

Cf. A168452, A004304, A005568, A000108, variant: A168357.

{a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)*(binomial(2*m,m)/(m+1))));polcoeff((x/serreverse(x*Ser(C_2)^2)),n)}

G.f.: A(x) = x/Series_Reversion(x*F(x)^2) where F(x) = g.f. of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).

G.f.: A(x) = F(x/A(x))^2 where A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A005568.

A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 2, 6, 2, 5, 30, 30, 5, 14, 140, 280, 140, 14, 42, 630, 2100, 2100, 630, 42, 132, 2772, 13860, 23100, 13860, 2772, 132, 429, 12012, 84084, 210210, 210210, 84084, 12012, 429, 1430, 51480, 480480, 1681680, 2522520, 1681680, 480480, 51480, 1430

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 1 - k.

A tree-rooted planar map is a planar map with a distinguished spanning tree.

			Triangle begins:
    1;
    1,     1;
    2,     6,     2;
    5,    30,    30,      5;
   14,   140,   280,    140,     14;
   42,   630,  2100,   2100,    630,    42;
  132,  2772, 13860,  23100,  13860,  2772,   132;
  429, 12012, 84084, 210210, 210210, 84084, 12012, 429;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
R. C. Mullin, On the Enumeration of Tree-Rooted Maps, Canadian Journal of Mathematics, Volume 19, 1967, pp. 174-183.

Columns k=0..2 are A000108, A002457, 2*A002803.
Row sums are A005568.
Central coefficients are A342983.
Cf. A001263, A215288, A269920, A342984.

Table[(2 n)!/(k!*(k + 1)!*(n - k)!*(n - k + 1)!), {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)

T(n,k) = {(2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!)}
{ for(n=0, 10, print(vector(n+1, k, T(n,k-1)))) }

T(n,k) = (2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!).
T(n,n-k) = T(n,k).
T(n, floor(n/2)) = A215288(n).
T(n,k) = A000108(n) * A001263(n+1,k+1). - Werner Schulte, Apr 04 2021

cp's OEIS Frontend

A168452 Self-convolution of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).

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A005817 a(n) = C(floor(n/2 + 1/2))*C(floor(n/2 + 1)) where C(i) = Catalan numbers A000108.

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A064037 Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant.

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A000356 Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).

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A004304 Number of nonseparable planar tree-rooted maps with n edges.

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A067640 Table T(n,k) giving number of two-legged knot diagrams with n >= 0 self-intersections and k >= 0 tangencies, read by antidiagonals.

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A172392 a(n) = C(2n,n)*C(2n+2,n+1)/(n+2).

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