A143547 - OEIS

A143547

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

1, 1, 1, 4, 7, 34, 70, 368, 819, 4495, 10472, 59052, 141778, 814506, 1997688, 11633440, 28989675, 170574723, 430321633, 2552698720, 6503352856, 38832808586, 99726673130, 598724403680, 1547847846090, 9335085772194, 24269405074740, 146936230074004, 383846168712104

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Number of achiral noncrossing partitions composed of n blocks of size 7. - Andrew Howroyd, Feb 08 2024

Number of achiral polyominoes composed of n octagonal cells of the hyperbolic regular tiling with Schläfli symbol {8,oo}. - Robert A. Russell, Oct 15 2025

LINKS

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. - From N. J. A. Sloane, Jul 12 2011

FORMULA

G.f.: A(x) = G(x^2) + x*G(x^2)^4 where G(x^2) = A(x)*A(-x) and G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

a(2n) = binomial(7*n,n)/(6*n+1); a(2n+1) = binomial(7*n+3,n)*4/(6*n+4).

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

a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+2*j+2*k+2*l=n-1} a(i) * a(2*j) * a(2*k) * a(2*l). - Seiichi Manyama, Jul 07 2025

a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} (-1)^(x_1+x_2+x_3) * Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

a(n) ~ c * 7^(7*n/2) / (2^(3*n+1/2) * 3^(3*n+3/2) * n^(3/2) * sqrt(Pi)), where c = 4 if n is odd and c = sqrt(7) if n is even. - Amiram Eldar, Sep 16 2025

From Robert A. Russell, Oct 21 2025: (Start)

a(n) = 2*A389937(n) - A389936(n) = A389936(n) - 2*A389938(n) = A389937(n) - A389938(n).

a(2m) = A002296(m) ~ (7^7/6^6)^m * sqrt(7/(2*Pi*(6*m)^3)).

a(2m+1) = A386392(m) ~ 4 * (7^7/6^6)^m * (7/6)^3 * sqrt(7/(2*Pi*(6*m)^3)).

a(n+2)/a(n) ~ 8^8/7^7; a(2m+1)/a(2m) ~ 4*8^3/7^3; a(2m)/a(2m-1) ~ 2*8^4/7^4. (End)

D-finite with recurrence: -96889010407*(7*n + 4)*(7*n + 1)*(7*n + 5)*(7*n + 2)*(7*n + 6)*(7*n + 3)*a(n) - 1210608210*(386476965*n^6 + 3864769650*n^5 + 15774680446*n^4 + 33749118676*n^3 + 40003218437*n^2 + 24945687242*n + 6400904424)*a(n + 1) + 9680832*(129048835153*n^6 + 2193096420788*n^5 + 15321406832655*n^4 + 56395838799300*n^3 + 115481221863532*n^2 + 124849959789352*n + 55723036859920)*a(n + 2) - 24192*(61252893009*n^6 + 13522353585741*n^5 + 194085830094450*n^4 + 1169438115914740*n^3 + 3631214352405976*n^2 + 5787541470553024*n + 3776998834027200)*a(n + 3) - 1381847040*(n + 4)*(3*n + 11)*(3*n + 13)*(5688018*n^3 + 45504144*n^2 + 120798629*n + 107102517)*a(n + 4) + 19575346692096*(3*n + 11)*(3*n + 16)*(3*n + 13)*(n + 5)*(n + 4)*(3*n + 14)*a(n + 5) = 0. - Robert Israel, Mar 19 2026

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 7*x^4 + 34*x^5 + 70*x^6 + 368*x^7 + ...

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then

A(x)*A(-x) = G(x^2) and A(x) = G(x^2) + x*G(x^2)^4 where

G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 + ...

G(x)^4 = 1 + 4*x + 34*x^2 + 368*x^3 + 4495*x^4 + 59052*x^5 + ...

form the bisections of A(x).

By definition, A(x) = 1 + x*A(x)^4*A(-x)^3 where

A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 95*x^4 + 332*x^5 + 1074*x^6 + ...

A(-x)^3 = 1 - 3*x + 6*x^2 - 19*x^3 + 51*x^4 - 183*x^5 + 550*x^6 -+ ...

MAPLE

f:= gfun:-rectoproc({96889010407*(7*n + 4)*(7*n + 1)*(7*n + 5)*(7*n + 2)*(7*n + 6)*(7*n + 3)*a(n) - 1210608210*(386476965*n^6 + 3864769650*n^5 + 15774680446*n^4 + 33749118676*n^3 + 40003218437*n^2 + 24945687242*n + 6400904424)*a(n + 1) + 9680832*(129048835153*n^6 + 2193096420788*n^5 + 15321406832655*n^4 + 56395838799300*n^3 + 115481221863532*n^2 + 124849959789352*n + 55723036859920)*a(n + 2) - 24192*(61252893009*n^6 + 13522353585741*n^5 + 194085830094450*n^4 + 1169438115914740*n^3 + 3631214352405976*n^2 + 5787541470553024*n + 3776998834027200)*a(n + 3) - 1381847040*(n + 4)*(3*n + 11)*(3*n + 13)*(5688018*n^3 + 45504144*n^2 + 120798629*n + 107102517)*a(n + 4) + 19575346692096*(3*n + 11)*(3*n + 16)*(3*n + 13)*(n + 5)*(n + 4)*(3*n + 14)*a(n + 5), a(0)=1, a(1)=1, a(2)=1, a(3)=4, a(4)=7}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Mar 19 2026

MATHEMATICA

terms = 26;

A[_] = 1; Do[A[x_] = 1 + x A[x]^4 A[-x]^3 + O[x]^terms // Normal, {terms}];

CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)

Table[If[OddQ[n], 4Binomial[(7n-1)/2, 3n], Binomial[7n/2, 3n]]/(3n+1), {n, 0, 40}] (* Robert A. Russell, Oct 15 2025 *)

PROG

(PARI) {a(n)=my(A=1+O(x^(n+1))); for(i=0, n, A=1+x*A^4*subst(A^3, x, -x)); polcoef(A, n)}

(PARI) {a(n)=my(m=n\2, p=3*(n%2)+1); binomial(7*m+p-1, m)*p/(6*m+p)}

CROSSREFS

Column k=7 of A369929 and k=8 of A370062.

Cf. A143046, A143546, A143549, A143550, A143553.