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

A213335 G.f. satisfies: A(x) = 1 + x/A(-x)^4.

Original entry on oeis.org

1, 1, 4, -6, -84, 171, 2940, -6576, -124260, 291321, 5810120, -14012244, -289392508, 711239741, 15052561056, -37498302048, -808073773572, 2033589755205, 44436219882252, -112715767473482, -2490257138332712, 6356863001632326, 141706826771491368
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 - 6*x^3 - 84*x^4 + 171*x^5 + 2940*x^6 - 6576*x^7 +...
where
1/A(-x) = 1 + x - 3*x^2 - 13*x^3 + 77*x^4 + 402*x^5 - 2849*x^6 - 16040*x^7 +...
1/A(-x)^4 = 1 + 4*x - 6*x^2 - 84*x^3 + 171*x^4 + 2940*x^5 - 6576*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 28*x^3 - 263*x^4 - 476*x^5 + 8740*x^6 +...
The g.f. G(x) of A213336 begins:
G(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
where G(x) = A(x*G(x)^4) and G(x/A(x)^4) = A(x);
also, G(x) = F(x/(1-x)^4) where F(x) = 1 + x*F(x)^4 is g.f. of A002293:
F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Links

Crossrefs

Cf. A213336, A213252, A213281, A143047; A002293.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x/subst(A^4, x, -x+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = G(x/A(x)^4) where G(x) = A(x*G(x)^4) is the g.f. of A213336.
G.f. satisfies: A(x) = ( x/Series_Reversion( x*F(x/(1-x)^4)^4 ) )^(1/4) where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
G.f. satisfies: A(x) = A(x)*A(-x) + x/A(x)^3.

A143046 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^3.

Original entry on oeis.org

1, 1, -3, -6, 35, 87, -588, -1578, 11511, 32223, -245883, -706824, 5556564, 16267508, -130617600, -387533058, 3161190783, 9474886287, -78241316361, -236394953670, 1971270824859, 5994591989967, -50388913722480, -154052058035736
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 - 6*x^3 + 35*x^4 + 87*x^5 - 588*x^6 - 1578*x^7 +...
where
A(x)^3 = 1 + 3*x - 6*x^2 - 35*x^3 + 87*x^4 + 588*x^5 - 1578*x^6 - 11511*x^7 +...
A(x)^4 = 1 + 4*x - 6*x^2 - 56*x^3 + 87*x^4 + 1008*x^5 - 1578*x^6 - 20464*x^7 +...
Note that a bisection of A^4 equals a bisection of A^3.

Links

Crossrefs

Cf. A143045, A143047, A143048, A143049, A213252, A213281, A213335.
Cf. A171200.

Programs

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

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^3)^3.
G.f. satisfies: [A(x)^4 + A(-x)^4]/2 = [A(x)^3 + A(-x)^3]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k>=0 and i+j+k=n-1} a(i) * a(j) * a(k). - Seiichi Manyama, Jul 08 2025

A143048 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^5.

Original entry on oeis.org

1, 1, -5, -15, 165, 630, -8151, -33780, 474045, 2052495, -30206330, -134392230, 2040588775, 9248893360, -143569282680, -659546365020, 10407737293965, 48303692377425, -771991701692175, -3611789245335285, 58311219888996170, 274581478640096340
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			A(x) = 1 + x - 5*x^2 - 15*x^3 + 165*x^4 + 630*x^5 - 8151*x^6 -++-...
A(x)^5 = 1 + 5*x - 15*x^2 - 165*x^3 + 630*x^4 + 8151*x^5 - 33780*x^6 -...
A(x)^6 = 1 + 6*x - 15*x^2 - 220*x^3 + 630*x^4 + 11286*x^5 - 33780*x^6 -...
Note that a bisection of A^6 equals a bisection of A^5.

Links

Crossrefs

Cf. A143045, A143046, A143047, A143049, A213252, A213281, A213335.
Cf. A171204.

Programs

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

Formula

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

A143049 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^6.

Original entry on oeis.org

1, 1, -6, -21, 286, 1281, -20592, -100226, 1749462, 8899086, -162993402, -852079872, 16106878320, 85783258295, -1658113447608, -8950840125828, 175904428301062, 959332126312266, -19096256882857668, -104984591307499239, 2111233112316364434
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			A(x) = 1 + x - 6*x^2 - 21*x^3 + 286*x^4 + 1281*x^5 - 20592*x^6 -++-...
A(x)^6 = 1 + 6*x - 21*x^2 - 286*x^3 + 1281*x^4 + 20592*x^5 - 100226*x^6 -...
A(x)^7 = 1 + 7*x - 21*x^2 - 364*x^3 + 1281*x^4 + 27027*x^5 - 100226*x^6 -...
Note that a bisection of A^7 equals a bisection of A^6.

Links

Crossrefs

Cf. A143045, A143046, A143047, A143048, A213252, A213281, A213335.
Cf. A171206.

Programs

  • PARI
    a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^6);polcoeff(A,n)

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^6)^6.
G.f. satisfies: [A(x)^7 + A(-x)^7]/2 = [A(x)^6 + A(-x)^6]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025

A171202 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^4.

Original entry on oeis.org

1, 1, 8, 152, 5664, 399376, 53846016, 14141384704, 7330134466560, 7551251740344320, 15510852680588984320, 63626087316632048238592, 521607805205244557347782656, 8549156556447111748331767857152, 280190094729160875643888549840814080, 18364219805837823940403573170370661842944
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200, A171201, A171203, A171204-A171211, A343439.
Cf. A143047, A171193.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1 + x*A[2x]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^4); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.