A143554 -id:A143554 - cp's OEIS frontend

A143550 G.f. A(x) satisfies A(x) = 1 + xA(x)^4A(-x)^2.

1, 1, 2, 11, 38, 257, 1040, 7646, 33374, 256718, 1171454, 9270560, 43558064, 351490167, 1686018600, 13799914556, 67223728270, 556203232266, 2741975026412, 22880729474777, 113875773363274, 956800135969601

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

			G.f. A(x) = 1 + x + 2*x^2 + 11*x^3 + 38*x^4 + 257*x^5 + 1040*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 72*x^3 + 333*x^4 + 1936*x^5 + 9966*x^6 +...
A(-x)^2 = 1 - 2*x + 5*x^2 - 26*x^3 + 102*x^4 - 634*x^5 + 2867*x^6 -+...
A(x)^2*A(-x) = 1 + x + 5*x^2 + 14*x^3 + 102*x^4 + 348*x^5 + 2867*x^6 +...
A(x)*A(-x) = 1 + 3*x^2 + 58*x^4 + 1597*x^6 + 51406*x^8 + 1807747*x^10 +...
[A(x)*A(-x)]^6 = 1 + 18*x^2 + 483*x^4 + 15342*x^6 + 535161*x^8 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..370
Vaclav Kotesovec, Recurrence

Cf. A143338, A143549, A143551, A143552, A143553, A143554.

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

G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^6.
G.f. satisfies: 1 - 4*y + 6*y^2 - 4*y^3 + y^4 - 2*x*y^6 + 4*x*y^7 - x*y^8 - x*y^9 + x^2*y^12 = 0, where y=A(x). - Vaclav Kotesovec, Mar 25 2014
a(n) ~ c / (sqrt(Pi)*n^(3/2)*r^n), where r = sqrt(22444621 + 5142958*sqrt(19))/46656 = 0.143559867369277217..., c = sqrt((13 - 49/sqrt(19))/3)/3 = 0.255214437... if n is even, and c = sqrt((73 - 1/sqrt(19))/3)/15 = 0.328341701... if n is odd. - Vaclav Kotesovec, Mar 25 2014
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} (-1)^(x_1+x_2) * Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025

A143551 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x).

Original entry on oeis.org

1, 1, 4, 29, 196, 1781, 14000, 139234, 1176340, 12283166, 108258380, 1165438808, 10561185568, 116096795195, 1072964739264, 11975785105572, 112313638368948, 1268177365551626, 12029082865935512, 137067430786661911

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 29*x^3 + 196*x^4 + 1781*x^5 + 14000*x^6 +...
Related expansions:
A(x)^5 = 1 + 5*x + 30*x^2 + 235*x^3 + 1845*x^4 + 16576*x^5 + 144270*x^6 +...
A(x)*A(-x) = 1 + 7*x^2 + 350*x^4 + 25165*x^6 + 2121330*x^8 +...
[A(x)*A(-x)]^6 = 1 + 42*x^2 + 2835*x^4 + 231350*x^6 +...

Cf. A143550, A143552, A143553, A143554.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A,x,-x));polcoeff(A,n)}

G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^6.

a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} (-1)^x_1 * Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025

A143553 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^3.

Original entry on oeis.org

1, 1, 2, 14, 50, 432, 1818, 17082, 77714, 763967, 3637718, 36786268, 180481258, 1860798032, 9324573430, 97502825964, 496344066386, 5245970686152, 27032002846992, 288124627083382, 1499144278319270, 16087838913122064

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

			G.f. A(x) = 1 + x + 2*x^2 + 14*x^3 + 50*x^4 + 432*x^5 + 1818*x^6 +...
Related expansions:
A(x)^5 = 1 + 5*x + 20*x^2 + 120*x^3 + 635*x^4 + 4301*x^5 + 25360*x^6 +...
A(-x)^3 = 1 - 3*x + 9*x^2 - 55*x^3 + 252*x^4 - 1818*x^5 + 9560*x^6 -+...
A(x)*A(-x) = 1 + 3*x^2 + 76*x^4 + 2776*x^6 + 118940*x^8 +...
[A(x)*A(-x)]^8 = 1 + 24*x^2 + 860*x^4 + 36488*x^6 + 1700198*x^8 +...

Cf. A143551, A143552, A143554.
Cf. A143046, A143547.
Cf. A143338, A143550.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A^3,x,-x));polcoeff(A,n)}

G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^8.

a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} (-1)^(x_1+x_2+x_3) * Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 08 2025

A143552 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^2.

Original entry on oeis.org

1, 1, 3, 22, 115, 1048, 6418, 63784, 421195, 4386273, 30271136, 324599018, 2306033386, 25228297188, 182938978344, 2030788315648, 14952369357211, 167836915812087, 1250429798513035, 14158770843121424, 106483223789898776

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 22*x^3 + 115*x^4 + 1048*x^5 + 6418*x^6 +...
Related expansions:
A(x)^5 = 1 + 5*x + 25*x^2 + 180*x^3 + 1200*x^4 + 9851*x^5 + 73195*x^6 +...
A(-x)^2 = 1 - 2*x + 7*x^2 - 50*x^3 + 283*x^4 - 2458*x^5 + 16106*x^6 -+...
A(x)*A(-x) = 1 + 5*x^2 + 195*x^4 + 10946*x^6 + 720443*x^8 +...
[A(x)*A(-x)]^7 = 1 + 35*x^2 + 1890*x^4 + 121947*x^6 + 8674036*x^8 +...

Cf. A143550, A143551, A143553, A143554.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A^2,x,-x));polcoeff(A,n)}

G.f. satisfies: A(x) + A(-x) = 1 + [A(x)*A(-x)] + x^2*[A(x)*A(-x)]^7.

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) * Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2.

Original entry on oeis.org

1, 1, 4, 8, 28, 80, 308, 984, 3980, 13472, 56164, 197032, 838396, 3013872, 13015188, 47624568, 207971436, 771336512, 3397886660, 12736715592, 56502898140, 213618833808, 953139545076, 3629043226392, 16270547827020, 62317467147744

Offset: 0

Paul D. Hanna, Aug 24 2008

nonn

Specific values:  A(2/9) = 17/9  and  A(-2/9) = 17/18.
Radius of convergence: r = sqrt(2*sqrt(3)-3)/3 = 0.2270833462...
with A(r) = (2 + sqrt(1-3*r))*(1+r^2)/(1+r) = 2.19775350...
and A(-r) = (2 - sqrt(1+3*r))*(1+r^2)/(1-r) = 3*(1+r^2) - A(r) = 0.9569470...
At x=r, the equation (*) (1+x^2)^2 - 2*(1+x^2)*y + (1+x)*y^2 - x*y^3 = 0, which is satisfied by y = A(x), factors out to:  (y - A(r))^2 * (y - A(r)*(1+r^2)/(2*(A(r)-1-r^2))) = 0; this gives the relation: (A(r)-1-r^2)*(3+3*r^2-A(r)) = r*A(r)^2.  At x>r, the equation (*) admits complex solutions for y.

			G.f. A(x) = 1 + x + 4*x^2 + 8*x^3 + 28*x^4 + 80*x^5 + 308*x^6 +...
A(x)/A(-x) = 1 + 2*x + 2*x^2 + 10*x^3 + 18*x^4 + 98*x^5 + 210*x^6 +...
where 1 - (1+x^2)/A(x) = x*A(x)/A(-x).
Related expansions:
A(x)^2/A(-x)^2 = 1 + 4*x + 8*x^2 + 28*x^3 + 80*x^4 + 308*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 24*x^3 + 88*x^4 + 280*x^5 + 1064*x^6 +...
where A(x)^2/A(-x)^2 = A(x)^2 + x + x*A(-x).

Cf. A143339, A143554, A143556, A143557, A143558, A143559.

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

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) (1+x^2)^2 - 2*(1+x^2)*A(x) + (1+x)*A(x)^2 - x*A(x)^3 = 0.
(2) A(x) = 1 + x*A(x)^2 + x^2 + x^2*A(-x).
(3) A(x) = 1 + x^2 + x*A(x)^2/A(-x).
(4) A(x) = 1 + x^2/(1 - A(-x)).
(5) A(x) = 1 + ( 1 - (1+x^2)/A(x) )^2/x.
(6) A(x) = (1+x^2)*G(x) where G(x) = 1 + x*G(x)^2/G(-x) is the g.f. of A143339.
Recurrence: (n-1)*(n+1)*(4*n^3 - 32*n^2 + 71*n - 30)*a(n) = 6*(8*n^3 - 56*n^2 + 101*n - 10)*a(n-1) + 6*(12*n^5 - 132*n^4 + 499*n^3 - 700*n^2 + 102*n + 305)*a(n-2) - 18*(n-4)*(8*n - 25)*a(n-3) + 27*(n-5)*(n-4)*(4*n^3 - 20*n^2 + 19*n + 13)*a(n-4). - Vaclav Kotesovec, Dec 29 2013
a(n) ~ c * 3^(n-1) * 2*sqrt(6*sqrt(3)-6 + sqrt(9+6*sqrt(3))) / (2*sqrt(Pi) * (2*sqrt(3)-3)^(n/2+1/4) * n^(3/2)), where c = 4/(2+12^(1/4)) if n is even and c = 12/(6+12^(3/4)) if n is odd. - Vaclav Kotesovec, Dec 29 2013

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

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

cp's OEIS Frontend

A143550 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x)^2.

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A143551 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x).

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A143553 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x)^3.

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A143552 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x)^2.

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A143555 G.f. satisfies: A(x) = 1 + x*A(x)^2/A(-x)^2.

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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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A143550 G.f. A(x) satisfies A(x) = 1 + xA(x)^4A(-x)^2.

A143551 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x).

A143553 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^3.

A143552 G.f. A(x) satisfies A(x) = 1 + xA(x)^5A(-x)^2.

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