A143547 -id:A143547 - cp's OEIS frontend

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

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

A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

Original entry on oeis.org

1, 1, 1, 7, 13, 112, 247, 2310, 5525, 53998, 135408, 1360289, 3518515, 36017352, 95223414, 988172368, 2655417765, 27844071255, 75769712590, 801012669457, 2201663313200, 23428926096576, 64924369564353, 694644371065372, 1938034271677595, 20829931845958872, 58448142042957576

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, A192893, A192894. From this one should be able to guess a g.f.

Andrew Howroyd, Table of n, a(n) for n = 0..200
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=13 of A369929 and k=14 of A370062.

Cf. A143049.

From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x)*A(-x))^6 ).
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^13.
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_7>=0 and x_1+2*(x_2+x_3+...+x_7)=n-1} a(x_1) * Product_{k=2..7} a(2*x_k). (End)
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_13>=0 and x_1+x_2+...+x_13=n-1} (-1)^(x_1+x_2+x_3+x_4+x_5+x_6) * Product_{k=1..13} a(x_k). - Seiichi Manyama, Jul 09 2025

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

a(0)=1 prepended by Seiichi Manyama, Jul 07 2025

A385688 E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^3 ).

Original entry on oeis.org

1, 1, 1, 10, 37, 736, 4861, 145552, 1392553, 55772416, 700205401, 35139710464, 546584937229, 32977620613120, 612127803448981, 43150087404292096, 930914421449463505, 75083676142358560768, 1846230024226716759601, 167681514857730519728128, 4629062510444281987051381

Offset: 0

Seiichi Manyama, Jul 06 2025

nonn

Cf. A058014, A385687.

Cf. A143547, A360988.

terms = 21; A[] = 1; Do[A[x] = Exp[x*((A[x] + A[-x])/2)^3] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 07 2025 *)

E.g.f. A(x) satisfies A(-x) = 1/A(x).

a(0) = 1; a(n) = (n-1)! * Sum_{i, j, k, l>=0 and i+2*j+2*k+2*l=n-1} (n-i) * a(i) * a(2*j) * a(2*k) * a(2*l)/(i! * (2*j)! * (2*k)! * (2*l)!).

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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).

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A192894 Number of symmetric 13-ary factorizations of the n-cycle (1,2...n).

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A385688 E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^3 ).

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