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.

A143547 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4*A(-x)^3.

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2008

Keywords

Comments

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

Examples

			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 -+ ...

Links

Crossrefs

Column k=7 of A369929 and k=8 of A370062.
Cf. A002296 (bisection), A143546.
Cf. A143549, A143550.
Cf. A143046, A143553.

Programs

  • 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 *)
  • 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)}

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

Extensions

a(26) onwards from Andrew Howroyd, Feb 08 2024

A143045 G.f.: A(x) = x + A(-x)^2.

Original entry on oeis.org

1, 1, -2, -3, 10, 18, -68, -131, 530, 1062, -4476, -9198, 39844, 83332, -368136, -780003, 3497058, 7483806, -33940940, -73210874, 335103340, 727473084, -3355045304, -7322240718, 33982884884, 74498594492, -347600543192, -764936992764, 3585459509640, 7916276980872, -37253166379536
Offset: 1

Views

Author

Paul D. Hanna, Jul 19 2008, Jul 20 2008

Keywords

Examples

			A(x) = x + x^2 - 2*x^3 - 3*x^4 + 10*x^5 + 18*x^6 - 68*x^7 - 131*x^8 +...
A(x)^2 = x^2 + 2*x^3 - 3*x^4 - 10*x^5 + 18*x^6 + 68*x^7 - 131*x^8 - 530*x^9 +...
A(x)^3 = x^3 + 3*x^4 - 3*x^5 - 20*x^6 + 18*x^7 + 153*x^8 - 131*x^9 -++-...
Note that a bisection of A^3 equals a bisection of A.

Links

Crossrefs

Cf. A143046, A143048, A143049, A213252, A100238 (related by series reversion).

Programs

  • Mathematica
    Rest[CoefficientList[1 + InverseSeries[Series[(Sqrt[1+4*x-4*x^2]-1)/2 + x^2, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Dec 27 2013 *)
  • PARI
    {a(n)=if(n<1,0,polcoeff(serreverse((sqrt(1+4*x-4*x^2 +x*O(x^n))-1)/2 + x^2),n))}
  • PARI
    {a(n)=local(A=x+x^2);for(i=0,n,A=x+subst(A,x,(-x+x*O(x^n)))^2);polcoeff(A,n)}
  • PARI
    {a(n)=local(A=x+x*O(x^n));for(i=0,n,A = x + x^2 - 2*x*A^2 + A^4);polcoeff(A,n)}

Formula

G.f.: A(x) = Series_Reversion( (sqrt(1+4*x-4*x^2)-1)/2 + x^2 ).
G.f. satisfies: A(x) = x + ( x - A(x)^2 )^2.
G.f. satisfies: [A(x)^3 - A(-x)^3]/2 = x*[A(x) + A(-x)]/2.
Recurrence: 27*(n-2)*(n-1)*n*(16*n^3 - 160*n^2 + 521*n - 552)*a(n) = 18*(n-2)*(n-1)*(32*n - 105)*a(n-1) - 12*(n-2)*(384*n^5 - 5376*n^4 + 29288*n^3 - 77560*n^2 + 99709*n - 49665)*a(n-2) + 48*(2*n - 7)*(32*n^3 - 248*n^2 + 623*n - 505)*a(n-3) + 64*(n-4)*(2*n - 9)*(2*n - 7)*(16*n^3 - 112*n^2 + 249*n - 175)*a(n-4). - Vaclav Kotesovec, Dec 27 2013
Limit n->infinity |a(n)|^(1/n) = 4/3*sqrt(3+2*sqrt(3)) = 3.3899463424498833... - Vaclav Kotesovec, Dec 27 2013

A143047 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^4.

Original entry on oeis.org

1, 1, -4, -10, 84, 265, -2604, -8900, 94692, 337940, -3767312, -13812674, 158785964, 593029550, -6967201736, -26372738120, 314904180100, 1204230041900, -14560722724912, -56130528427400, 685514219386576, 2659770565898729, -32749512944380172
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			A(x) = 1 + x - 4*x^2 - 10*x^3 + 84*x^4 + 265*x^5 - 2604*x^6 - 8900*x^7 +...
A(x)^4 = 1 + 4*x - 10*x^2 - 84*x^3 + 265*x^4 + 2604*x^5 - 8900*x^6 -...
A(x)^5 = 1 + 5*x - 10*x^2 - 120*x^3 + 265*x^4 + 3906*x^5 - 8900*x^6 -...
Note that a bisection of A^5 equals a bisection of A^4.

Links

Crossrefs

Cf. A143045, A143046, A143048, A143049, A213252, A213281, A213335.
Cf. A171202.

Programs

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

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^4)^4.
G.f. satisfies: [A(x)^5 + A(-x)^5]/2 = [A(x)^4 + A(-x)^4]/2.
a(0) = 1; a(n) = (-1)^(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

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

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
Showing 1-6 of 6 results.

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