cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A143554 G.f. A(x) satisfies A(x) = 1 + x*A(x)^5*A(-x)^4.

Original entry on oeis.org

1, 1, 1, 5, 9, 55, 117, 775, 1785, 12350, 29799, 211876, 527085, 3818430, 9706503, 71282640, 184138713, 1366368375, 3573805950, 26735839650, 70625252863, 531838637759, 1416298046436, 10723307329700, 28748759731965, 218658647805780, 589546754316126
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Comments

Number of achiral noncrossing partitions composed of n blocks of size 9. - Andrew Howroyd, Feb 08 2024

Examples

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then
G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...

Links

Crossrefs

Column k=9 of A369929 and k=10 of A370062.
Cf. A143338, A143546, A143547, A143550, A062994 (bisection).
Cf. A143551, A143552, A143553.
Cf. A143047.

Programs

  • Mathematica
    terms = 25;
A[] = 1; Do[A[x] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)
  • PARI
    {a(n)=my(A=1+x*O(x^n));for(i=0,n,A=1+x*A^5*subst(A^4,x,-x));polcoef(A,n)}
  • PARI
    {a(n)=my(m=n\2,p=4*(n%2)+1);binomial(9*m+p-1,m)*p/(8*m+p)}

Formula

G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.
a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+2*j+2*k+2*l+2*m=n-1} a(i) * a(2*j) * a(2*k) * a(2*l) * a(2*m). - Seiichi Manyama, Jul 07 2025
a(0) = 1; a(n) = Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (-1)^(x_1+x_2+x_3+x_4) * Product_{k=1..9} a(x_k). - Seiichi Manyama, Jul 09 2025

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

Original entry on oeis.org

1, 1, 2, 11, 38, 257, 1040, 7646, 33374, 256718, 1171454, 9270560, 43558064, 351490167, 1686018600, 13799914556, 67223728270, 556203232266, 2741975026412, 22880729474777, 113875773363274, 956800135969601
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

  • PARI
    {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)}

Formula

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

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

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			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 +...

Crossrefs

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

Programs

  • PARI
    {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)}

Formula

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 + x*A(x)^5*A(-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

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			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 +...

Crossrefs

Cf. A143550, A143551, A143553, A143554.

Programs

  • PARI
    {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)}

Formula

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

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

Views

Author

Paul D. Hanna, Aug 09 2008

Keywords

Examples

			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 +...

Crossrefs

Cf. A047749, A143549, A143551.
Cf. A143546.

Programs

  • PARI
    {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)}

Formula

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 + x*A(x)^4*A(-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

Views

Author

Paul D. Hanna, Aug 24 2008

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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)}

Formula

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
Showing 1-6 of 6 results.

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