A047749 -id:A047749 - cp's OEIS frontend

A385617 G.f. A(x) satisfies A(x) = 1/( 1 - x(A(x) + A(2x)) ).

1, 2, 10, 82, 1062, 22646, 846570, 58644858, 7808479582, 2038568219422, 1054007965984050, 1084591195956246130, 2226674324358059364150, 9131600163886719149539590, 74851744440590132840318820090, 1226745312860243142951267683147178, 40204124737879503807503331117931168974

Offset: 0

Seiichi Manyama, Jul 05 2025

nonn

Cf. A000051, A015083, A047749, A385618, A385622.

terms = 17; A[] = 1; Do[A[x] = 1/( 1 - x*(A[x] + A[2*x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 05 2025 *)

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2^j+1)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} (2^k+1) * a(k) * a(n-1-k).

a(n) ~ c * 2^(n*(n-1)/2), where c = 30.250837358072598377515060923766952434821313428993180484... - Vaclav Kotesovec, Jul 05 2025

A065942 Central column of triangle A065941.

Original entry on oeis.org

1, 1, 3, 4, 15, 21, 84, 120, 495, 715, 3003, 4368, 18564, 27132, 116280, 170544, 735471, 1081575, 4686825, 6906900, 30045015, 44352165, 193536720, 286097760, 1251677700, 1852482996, 8122425444, 12033222880, 52860229080, 78378960360

Offset: 0

Len Smiley, Nov 29 2001

When viewed as (1,1), (3,4), (15,21), ... this represents a shallow staircase on Pascal's triangle, arranged as a square array. - Paul Barry, Mar 11 2003
Also central column of triangle A011973 (taking rows with odd number of terms only). - John Molokach, Jul 08 2013
Interleaving of A005809 and A045721. - Bruce J. Nicholson, Apr 24 2018

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 15*x^4 + 21*x^5 + 84*x^6 + 120*x^7 + ... - _Michael Somos_, Jun 23 2018

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)

Michael De Vlieger, Table of n, a(n) for n = 0..2416
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271; Corrections.

Cf. A065941 (complete triangle), A047749.

Cf. A005809, A045721.

List([0..40],n->Binomial(n+Int(n/2),n)); # Muniru A Asiru, Apr 28 2018

Array[Binomial[# + Floor[#/2], #] &, 30, 0] (* Michael De Vlieger, Apr 27 2018 *)

a(n) = binomial(n+n\2, n); \\ Altug Alkan, Apr 24 2018

a(n) = binomial(2n-floor((n+1)/2), floor(n/2)).
a(n+1) = Sum_{k=0..ceiling(n/2)} binomial(n+k, k). - Benoit Cloitre, Mar 06 2004
a(n) = binomial(n+floor(n/2), n). - Paul Barry, May 18 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+k, k). - Paul Barry, Jul 06 2004
a(2n-1) = binomial(3n-3,n-1); a(2n) = binomial(3n-2,n-1). - John Molokach, Jul 08 2013
G.f.: A(x) = x*(d/dx)[log(S(x)-1)] = x*[(d/dx) S(x)]/[S(x)-1], where S(x) is the g.f. of A047749. - Vladimir Kruchinin, Jun 12 2014.
Conjecture: 8*n*(n-1)*a(n) -36*(n-1)*(n-3)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jun 13 2014
0 = a(n)*(+281138850*a(n+2) +729089100*a(n+3) -77071527*a(n+4) -134472793*a(n+5)) +a(n+1)*(+15618825*a(n+2) -1650969*a(n+3) -9342280*a(n+4) -1729448*a(n+5)) +a(n+2)*(-19089675*a(n+2) -61394833*a(n+3) +6470716*a(n+4) +14929796*a(n+5)) +a(n+3)*(-1291668*a(n+3) +553572*a(n+4) +246032*a(n+5)) for all n in Z. - Michael Somos, Jun 23 2018

A385149 Number of chiral pairs of asymmetric polyominoes with n cells of the regular tiling with Schläfli symbol {4,oo}.

Original entry on oeis.org

0, 0, 0, 0, 1, 8, 43, 225, 1162, 6081, 32315, 174856, 961764, 5369567, 30373643, 173811011, 1004802212, 5861460314, 34468644574, 204161097084, 1217143092549, 7299002607829, 44005589820244, 266608357403244, 1622502342468552, 9914884364399700

Offset: 0

Robert A. Russell, Jun 19 2025

A stereographic projection of the {4,oo} tiling on the Poincaré disk can be obtained via the Christensson link. Each member of a chiral pair is a reflection but not a rotation of the other.

			 __ __ __    __ __ __
|__|__|__|  |__|__|__|  a(4) = 1.
      |__|  |__|

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Robert A. Russell, Stereographic projections of chiral pairs of asymmetric polyominoes with 4 or 5 cells

Cf. A005034 (oriented), A005036 (unoriented), A369315 (chiral), A047749 (achiral), A001764 (rooted).

Table[If[n<4,0,(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1) + Switch[Mod[n,4], 0,4Binomial[3n/4,n/4]/(n/2+1)-6Binomial[3n/2,n/2]/(n+1), 1,(4Binomial[(3n-3)/4,(n-1)/4]-10Binomial[(3n-1)/2,(n-1)/2])/(n+1)+(8Binomial[(3n+1)/4,(n-1)/4]+16Binomial[(3n-3)/4,(n-5)/4])/(n+3), 2,16Binomial[(3n-2)/4,(n-2)/4]/(n+2)-6Binomial[3n/2,n/2]/(n+1), 3,24Binomial[(3n-1)/4,(n-3)/4]/(n+3)-10Binomial[(3n-1)/2,(n-1)/2]/(n+1)])/8],{n,0,30}]

G.f.: (3*G(z) - G(z)^2 - 6*G(z^2) - 5z*G(z^2)^2 + 4*G(z^4) + 2z*G(z^4) + 2z*G(z^4)^2 + 4z^2*G(z^4)^2 + 4z^3*G(z^4)^3 + 2z^5*G(z^4)^4) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764.

A047750 If n mod 2 = 0 then m := n/2 and a(n) = (3m)!(5m+1)/((m+1)!(2m+1)!); otherwise m := (n-1)/2, a(n) = 6(3m+2)!/(m!(2*m+3)!).

Original entry on oeis.org

1, 2, 3, 6, 11, 24, 48, 110, 231, 546, 1183, 2856, 6324, 15504, 34884, 86526, 197087, 493350, 1134705, 2861430, 6633315, 16829280, 39268320, 100134216, 234930276, 601661144, 1418201268, 3645533040, 8627761528, 22249511328

Offset: 0

N. J. A. Sloane

nonn

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.

Cf. A001764, A047749, A047760, A047773.

series(RootOf(x*A^3-2*A^2+3*A-1, A)^2, x=0, 30);  # Mark van Hoeij, May 16 2013

a[0] = 1; a[1] = 2; a[n_] := a[n] = 3(2n+3)(3n-4)(3n-2)a[n-2]/(4n(n+2)(2n+1)) + (3(18n+16)a[n-1])/(4n(n+2)(2n+1)); Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Dec 02 2016 *)
Table[If[OddQ[n],6Binomial[(3n+1)/2,n+1]/(n+2),(5n+2)Binomial[3n/2,n/2] / ((n+1)(n+2))],{n,0,30}] (* Robert A. Russell, Feb 16 2024 *)

a047750(n)={if(n%2,my(m=(n-1)/2);6*(3*m+2)!/(m!*(2*m+3)!),my(m=n/2);(3*m)!*(5*m+1)/((m+1)!*(2*m+1)!))};
for(k=0,29,print1(a047750(k),", ")) \\ Hugo Pfoertner, Mar 07 2020

From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = sum of top row terms in M^n, M = the infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  0, 0, 1, 0, 0, 0, ...
  1, 1, 0, 1, 0, 0, ...
  0, 0, 1, 0, 1, 0, ...
  1, 1, 0, 1, 0, 1, ...
  ... (End)
8*n*(n+2)*a(n) + 4*(7*n^2 - 7*n - 17)*a(n-1) + 6*(-9*n^2 + 9*n - 17)*a(n-2) - 21*(3*n-5)*(3*n-7)*a(n-3) = 0. - R. J. Mathar, Jul 10 2013
From Robert A. Russell, Mar 20 2024: (Start)
a(n) = V(n) in the Beineke and Pippert link.
G.f.: 2*(G(z^2) - 1)/z + 2*G(z^2)^2 - G(z^2), where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

A124305 Riordan array (1, 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/3).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 7, 0, 4, 0, 1, 0, 12, 0, 12, 0, 5, 0, 1, 0, 0, 30, 0, 18, 0, 6, 0, 1, 0, 55, 0, 55, 0, 25, 0, 7, 0, 1, 0, 0, 143, 0, 88, 0, 33, 0, 8, 0, 1

Offset: 0

Paul Barry, Oct 25 2006

			Triangle begins
  1,
  0,  1,
  0,  0,  1,
  0,  1,  0,  1,
  0,  0,  2,  0,  1,
  0,  3,  0,  3,  0,  1,
  0,  0,  7,  0,  4,  0,  1,
  0, 12,  0, 12,  0,  5,  0,  1
From _Paul Barry_, Sep 28 2009: (Start)
Production matrix is
  0, 1,
  0, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1,
  0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 (End)

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001477, A001764, A006013, A055998.

Cf. A047749 (row sums), A098746 (diagonal sums), A124304 (inverse).

A124305:= func< n,k | n eq 0 select 1 else (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial(n + Floor((n-k)/2) -1, n-1) >;
[A124305(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 25 2023

A124305[n_, k_]:= If[n==0, 1, (1/2)*(1+(-1)^(n-k))*(k/n)*Binomial[n +(n-k)/2 -1, (n-k)/2]];
Table[A124305[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 19 2023 *)

def A124305(n,k): return 1 if n==0 else ((n-k+1)%2)*k*binomial(n + (n-k)//2 -1, n-1)//n
flatten([[A124305(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 25 2023

Sum_{k=0..n} T(n, k) = A047749(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*(1 + (-1)^n)*A098746(n/2).
From G. C. Greubel, Aug 19 2023: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n-k))*(k/n)*binomial(n + (n-k)/2 - 1, (n-k)/2), with T(0, 0) = 1.
T(n, n) = 1.
T(n, n-2) = A001477(n-2).
T(n, n-4) = A055998(n-4).
T(n, n-6) = A111396(n-6).
T(n, 0) = 0^n.
T(n, 1) = ((1-(-1)^n)/2)*A001764(floor((n-1)/2)).
T(n, 2) = ((1+(-1)^n)/2)*A006013(floor((n-2)/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n * A047749(n). (End)

A143338 G.f. A(x) satisfies A(x) = 1 + xA(x)^3A(-x).

Original entry on oeis.org

1, 1, 2, 8, 26, 127, 478, 2536, 10250, 56900, 239880, 1370272, 5940054, 34607146, 153018932, 904441648, 4058644842, 24254529036, 110096276440, 663665021280, 3040205250984, 18455364854839, 85176971647470, 520059936017128

Offset: 0

Paul D. Hanna, Aug 09 2008

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 127*x^5 + 478*x^6 +...
Compare bisections of A(x)^2, A(x)^2*A(-x), and A(x)^4*A(-x)^2:
A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 72*x^4 + 338*x^5 + 1378*x^6 + 6952*x^7 +...
A(x)^2*A(-x) = 1 + x + 5*x^2 + 11*x^3 + 72*x^4 + 191*x^5 + 1378*x^6 + 3979*x^7 +...
A(x)^4*A(-x)^2 = 1 + 2*x + 11*x^2 + 32*x^3 + 191*x^4 + 636*x^5 + 3979*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 9*x^2 + 37*x^3 + 144*x^4 + 669*x^5 + 2882*x^6 + 14229*x^7 +...
A(x)^3*A(-x) = 1 + 2*x + 8*x^2 + 26*x^3 + 127*x^4 + 478*x^5 + 2536*x^6 +...
A(x)^3*A(-x)^2 = 1 + x + 8*x^2 + 14*x^3 + 127*x^4 + 264*x^5 + 2536*x^6 +...

Cf. A047749, A143549, A143551.

Cf. A143546.

{a(n)=local(A=1+x+O(x^21));for(i=0,n,A=1+x*A^3*subst(A,x,-x));polcoeff(A,n)}

a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (-1)^i * a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025

A143549 G.f. A(x) satisfies A(x) = 1 + xA(x)^4A(-x).

Original entry on oeis.org

1, 1, 3, 17, 85, 598, 3473, 26668, 166429, 1340079, 8724438, 72374714, 484498327, 4102336176, 28009706440, 240729330116, 1668007246157, 14499527706129, 101618389067849, 891275643857227, 6303425058175018, 55686806813191060

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 17*x^3 + 85*x^4 + 598*x^5 + 3473*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 18*x^2 + 108*x^3 + 635*x^4 + 4348*x^5 + 28336*x^6 +...
A(x)*A(-x) = 1 + 5*x^2 + 145*x^4 + 5971*x^6 + 287253*x^8 +...
[A(x)*A(-x)]^5 = 1 + 25*x^2 + 975*x^4 + 45605*x^6 + 2355490*x^8 +...

Robert Israel, Table of n, a(n) for n = 0..600

Cf. A143552, A143553, A143554.
Cf. A143550, A143547.
Cf. A047749, A143338, A143551.

S:= series(RootOf(_Z^15*x^3-_Z^12*x^2+_Z^11*x^2-_Z^4+4*_Z^3-6*_Z^2+4*_Z-1),x,31):
seq(coeff(S,x,i),i=0..30); # Robert Israel, Jul 10 2017

nmax = 21; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x*A[x]^4*A[-x]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

{a(n)=local(A=1+x*O(x^n));for(i=0,2*n,A=1+x*A^4*subst(A^1,x,-x));polcoeff(A,n)}

G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^5.
G.f. satisfies: -x^3*A(x)^15+x^2*A(x)^12-x^2*A(x)^11+A(x)^4-4*A(x)^3+6*A(x)^2-4*A(x)+1 = 0. - Robert Israel, Jul 10 2017
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (-1)^i * a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025

A192893 Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).

Original entry on oeis.org

1, 1, 1, 6, 11, 81, 176, 1406, 3311, 27636, 68211, 585162, 1489488, 13019909, 33870540, 300138696, 793542167, 7105216833, 19022318084, 171717015470, 464333035881, 4219267597578, 11502251937176, 105085831400550, 288417894029200, 2647012241261856, 7306488667126803

Offset: 0

N. J. A. Sloane, Jul 12 2011

nonn

The six sequences displayed in Table 1 of the Bousquet-Lamathe reference are A047749, A143546, A143547, A143554, this sequence, and A192894. From this one should be able to guess a g.f.

Number of achiral noncrossing partitions composed of n blocks of size 11. - Andrew Howroyd, Feb 08 2024

Andrew Howroyd, Table of n, a(n) for n = 0..500
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1.

Column k=11 of A369929 and k=12 of A370062.

Cf. A143048.

a(n)={my(m=n\2, p=5*(n%2)+1); binomial(11*m+p-1, m)*p/(10*m+p)} \\ Andrew Howroyd, Feb 08 2024

From Andrew Howroyd, Feb 08 2024: (Start)
a(2n) = binomial(11*n,n)/(10*n+1); a(2n+1) = binomial(11*n+5,n)*6/(10*n+6).
G.f. A(x) satisfies A(x) = 1 + x*A(x)^6*A(-x)^5. (End)
From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^11 is the g.f. of A230388.
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_6>=0 and x_1+2*(x_2+x_3+...+x_6)=n-1} a(x_1) * Product_{k=2..6} a(2*x_k). (End)
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_11>=0 and x_1+x_2+...+x_11=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5) * Product_{k=1..11} a(x_k). - Seiichi Manyama, Jul 09 2025

a(11) onwards from Andrew Howroyd, Jan 26 2024

a(0)=1 prepended by Andrew Howroyd, Feb 08 2024

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

Original entry on oeis.org

1, 3, 55, 1428, 43263, 1430715, 50067108, 1822766520, 68328754959, 2619631042665, 102240109897695, 4048514844039120, 162250238001816900, 6568517413771094628, 268225186597703313816, 11034966795189838872624, 456949965738717944767791

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=4, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

First bisection of A001764.

Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), this sequence (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

l:=4; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */

Table[Binomial[6 n, 2 n]/(4 n + 1), {n, 0, 20}]

a(n) = A047749(4*n-2) for n>0.
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 4F3(1/6,1/3,2/3,5/6; 1/2,3/4,5/4; 729*x/16).
a(n) ~ 3^(6*n+1/2)/(sqrt(Pi)*2^(4*n+7/2)*n^(3/2)). (End)

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).

Original entry on oeis.org

1, 12, 1428, 246675, 50067108, 11124755664, 2619631042665, 642312451217745, 162250238001816900, 41932353590942745504, 11034966795189838872624, 2946924270225408943665279, 796607831560617902288322405, 217550867863011281855594752680

Offset: 0

Bruno Berselli, Jan 12 2014

This is the case l=6, k=3 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.

Also, the sequence follows A002296 and A235536, namely binomial(7*n,n)/(6*n+1) and binomial(8*n,2*n)/(6*n+1); naturally, even binomial(10*n,4*n)/(6*n+1) is always integer.

Robert Israel, Table of n, a(n) for n = 0..403
Zhi-Wei Sun, On Divisibility Of Binomial Coefficients, Journal of the Australian Mathematical Society 93 (2012), p. 189-201.

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), A235536 (l=6, k=2), A187357 (l=3, k=3), this sequence (l=6, k=3).

l:=6; k:=3; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* here l is divisible by all the prime factors of k */

seq(binomial(9*n,3*n)/(6*n+1), n=0..30); # Robert Israel, Feb 15 2021

Table[Binomial[9 n, 3 n]/(6 n + 1), {n, 0, 20}]

a(n) = A001764(3*n) = A047749(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/9,2/9,4/9,5/9,7/9,8/9; 1/3,1/2,2/3,5/6,7/6; 19683*x/64).
a(n) ~ 3^(9*n-1)/(sqrt(Pi)*4^(3*n+1)*n^(3/2)). (End)
D-finite with recurrence 8*(6*n + 5)*(2*n + 1)*(n + 1)*(3*n + 2)*(3*n + 1)*(6*n + 7)*a(n + 1) = 3*(9*n + 8)*(9*n + 7)*(9*n + 5)*(9*n + 4)*(9*n + 2)*(9*n + 1)*a(n). - Robert Israel, Feb 15 2021

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A385617 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(2*x)) ).

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A065942 Central column of triangle A065941.

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A385149 Number of chiral pairs of asymmetric polyominoes with n cells of the regular tiling with Schläfli symbol {4,oo}.

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A047750 If n mod 2 = 0 then m := n/2 and a(n) = (3*m)!*(5*m+1)/((m+1)!*(2*m+1)!); otherwise m := (n-1)/2, a(n) = 6*(3*m+2)!/(m!*(2*m+3)!).

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A124305 Riordan array (1, 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/3).

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A143338 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x).

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A143549 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x).

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A192893 Number of symmetric 11-ary factorizations of the n-cycle (1,2...n).

A385617 G.f. A(x) satisfies A(x) = 1/( 1 - x(A(x) + A(2x)) ).

A047750 If n mod 2 = 0 then m := n/2 and a(n) = (3m)!(5m+1)/((m+1)!(2m+1)!); otherwise m := (n-1)/2, a(n) = 6(3m+2)!/(m!(2*m+3)!).

A124305 Riordan array (1, 2sqrt(3)sin(arcsin(3sqrt(3)x/2)/3)/3).

A143338 G.f. A(x) satisfies A(x) = 1 + xA(x)^3A(-x).

A143549 G.f. A(x) satisfies A(x) = 1 + xA(x)^4A(-x).

A235534 a(n) = binomial(6n, 2n) / (4*n + 1).

A235535 a(n) = binomial(9n, 3n) / (6*n + 1).