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 1-7 of 7 results.

A213252 G.f. satisfies: A(x) = 1 + x/A(-x)^2.

Original entry on oeis.org

1, 1, 2, -1, -10, 7, 88, -68, -946, 767, 11298, -9425, -144024, 122436, 1919440, -1653776, -26419778, 22992655, 372670246, -326863667, -5358911450, 4729547023, 78264621664, -69424933968, -1157715304760, 1031309398852, 17309542787288, -15474833826028
Offset: 0

Views

Author

Paul D. Hanna, Jun 07 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 - x^3 - 10*x^4 + 7*x^5 + 88*x^6 - 68*x^7 +...
where
x/A(-x)^2 = x + 2*x^2 - x^3 - 10*x^4 + 7*x^5 + 88*x^6 - 68*x^7 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 2*x^3 - 18*x^4 - 10*x^5 + 151*x^6 + 88*x^7 +...
The g.f. G(x) of A006319 begins:
G(x) = 1 + x + 4*x^2 + 16*x^3 + 68*x^4 + 304*x^5 + 1412*x^6 + 6752*x^7 +...
where G(x) = A(x*G(x)^2) and G(x/A(x)^2) = A(x);
also, G(x) = F(x/(1-x)^2) where F(x) = 1 + x*F(x)^2 is g.f. of A000108:
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...

Links

Crossrefs

Cf. A006319, A213281, A213335, A143045; A000108.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x/subst(A^2,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)^2) where G(x) = A(x*G(x)^2) is the g.f. of A006319 (royal paths in a lattice).
G.f. satisfies: A(x) = sqrt( x/Series_Reversion( x*C(x/(1-x)^2)^2 ) ) where C(x) = 1 + x*C(x)^2 = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).
G.f. satisfies: A(x) = A(x)*A(-x) + x/A(x).

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

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

A100238 G.f. A(x) satisfies: 2^n + 1 = Sum_{k=0..n} [x^k] A(x)^n for n>=1.

Original entry on oeis.org

1, 2, -2, 4, -12, 40, -144, 544, -2128, 8544, -35008, 145792, -615296, 2625792, -11311616, 49124352, -214838528, 945350144, -4182412288, 18593224704, -83015133184, 372090122240, -1673660915712, 7552262979584, -34178799378432, 155096251351040, -705533929816064
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2004

Keywords

Examples

			From the table of powers of A(x), we see that
2^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1: [1, 2], -2, 4, -12, 40, -144, 544, -2128, 8544, ...;
A^2: [1, 4, 0], 0, -4, 16, -64, 256, -1040, 4288, ...;
A^3: [1, 6, 6, -4], 0, 0, -8, 48, -240, 1120, -5088, ...;
A^4: [1, 8, 16, 0, -8], 0, 0, 0, -16, 128, -768, ...;
A^5: [1, 10, 30, 20, -20, -8], 0, 0, 0, 0, -32, ...;
A^6: [1, 12, 48, 64, -12, -48, 0], 0, 0, 0, 0, 0, ...;
A^7: [1, 14, 70, 140, 56, -112, -56, 16], 0, 0, 0, ...;
A^8: [1, 16, 96, 256, 240, -128, -256, 0, 32], 0, 0, ...; ...
In the above table of coefficients in A(x)^n, the main diagonal satisfies:
[x^n] A(x)^(n+1) = (n+1)*A009545(n+1) for n>=0.

Crossrefs

Cf. A100223, A071356, A009545, A143045.
a(n) = -(-1)^n * A025227(n), if n>1.

Programs

  • PARI
    {a(n)=if(n==0,1,(2^n+1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)}
  • PARI
    {a(n)=if(n==0,1,if(n==1,2,if(n==2,-2,(-2*(2*n-3)*a(n-1)+4*(n-3)*a(n-2))/n)))}
  • PARI
    {a(n)=polcoeff( (1+2*x+sqrt(1+4*x-4*x^2+x^2*O(x^n)))/2,n)}
  • PARI
    a(n)=polcoeff((1+2*x+sqrt(1+4*x-4*x^2+x*O(x^n)))/2,n)

Formula

a(n) = (-2*(2*n-3)*a(n-1) + 4*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=2, a(2)=-2.
G.f.: A(x) = (1+2*x + sqrt(1+4*x-4*x^2))/2.
G.f. satisfies: (2+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k] (A(x)+z*x)^n for all z, where [x^k] F(x) denotes the coefficient of x^k in F(x).
Given g.f. A(x), then B(x)=A(x)-1-x series reversion is -B(-x). - Michael Somos, Sep 07 2005
Given g.f. A(x) and C(x) = g.f. of A025225, then B(x)=A(x)-1-x satisfies B(x)=x-C(x*B(x)). - Michael Somos, Sep 07 2005
G.f.: 4x^2/(1+2x - sqrt(1+4x-4x^2)). - Michael Somos, Sep 08 2005

A216683 G.f. satisfies: A(x) = 1 + x*A(x) / ( A(I*x)*A(-I*x) ).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 2, 0, -5, -2, -2, 8, 38, 20, 20, -48, -269, -138, -138, 392, 2194, 1132, 1132, -3344, -19010, -9812, -9812, 30032, 172332, 89000, 89000, -279136, -1613629, -833626, -833626, 2663432, 15485978, 8002172, 8002172, -25938768, -151520246, -78309372, -78309372
Offset: 0

Views

Author

Paul D. Hanna, Sep 14 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 2*x^6 - 5*x^8 - 2*x^9 +...
Related expansions:
1/A(x) = 1 - x - x^3 + 2*x^5 + 3*x^7 - 10*x^9 - 18*x^11 + 68*x^13 + 131*x^15 - 530*x^17 - 1062*x^19 +...+ -A143045(n)*x^(2*n-1) +...
A(I*x)*A(-I*x) = 1 - x^2 + 3*x^4 - 2*x^6 - 5*x^8 + 2*x^10 + 38*x^12 - 20*x^14 - 269*x^16 + 138*x^18 + 2194*x^20 +...
The 4-sections of g.f. A(x) begin:
A0(x) = 1 + 3*x - 5*x^2 + 38*x^3 - 269*x^4 + 2194*x^5 - 19010*x^6 + 172332*x^7 +...
A1(x) = A2(x) = 1 + 2*x - 2*x^2 + 20*x^3 - 138*x^4 + 1132*x^5 - 9812*x^6 + 89000*x^7 +...
A3(x) = 2 + 8*x^2 - 48*x^3 + 392*x^4 - 3344*x^5 + 30032*x^6 - 279136*x^7 + 2663432*x^8 +...
where
A1(x) + x*A3(x)/(2*A0(x)) = 1 + 3*x - 5*x^2 + 38*x^3 - 269*x^4 + 2194*x^5 +...

Links

Crossrefs

Cf. A212527, A216681, A143045.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*A/(subst(A, x, I*x+x*O(x^n))*subst(A, x, -I*x+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: A(x) = 1/(1 - G(x^2)/x) where G(x) = x + G(-x)^2 is the g.f. of A143045.
a(4*n+1) = a(4*n+2) for n>=0.
Let A(x) = A0(x^4) + x*A1(x^4) + x^2*A2(x^2) + x^3*A3(x^4), then
(1) A1(x) = A2(x).
(2) A0(x) = A1(x) + x*A3(x) / (2*A0(x)).
(3) A0(x^4) - x^2*A2(x^4) = A(I*x)*A(-I*x).
Showing 1-7 of 7 results.

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