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.

A213282 G.f. satisfies A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 6, 36, 236, 1656, 12192, 92960, 727824, 5817696, 47281472, 389533056, 3245867136, 27308274688, 231654031104, 1979205694464, 17016094611712, 147104972637696, 1277988764697600, 11151534242977792, 97692088569096192, 858890594909048832, 7575804347863105536
Offset: 0

Views

Author

Paul D. Hanna, Jun 08 2012

Keywords

Comments

Compare to the g.f. B(x) of A006319 where B(x) = C(x/(1-x)^2) such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 36*x^3 + 236*x^4 + 1656*x^5 + 12192*x^6 +...
G.f.: A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Links

Crossrefs

Cf. A213281, A001764; variants: A006319 (royal paths in a lattice), A213336.

Programs

  • Maple
    series(RootOf(G = 1 + G^3*x/(1-x)^3, G),x=0,30); # Mark van Hoeij, Apr 18 2013
  • PARI
    /* G.f. A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3: */
{a(n)=local(A,G=1+x);for(i=1,n,G=1+x*G^3+x*O(x^n));A=subst(G,x,x/(1-x+x*O(x^n))^3);polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* G.f. A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3: */
{a(n)=local(F=1+x+x*O(x^n), A=1); for(i=1, n+1, F=1+x/subst(F^3, x, -x+x*O(x^n))); A=(serreverse(x/F^3)/x)^(1/3); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3 is the g.f. of A213281.
G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * binomial(3*k,k)/(2*k+1). - Seiichi Manyama, Oct 03 2023

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).

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

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
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