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 1-3 of 3 results.

A264225 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-6*x) ), with A(0) = 0.

Original entry on oeis.org

1, 3, 15, 81, 462, 2718, 16344, 99900, 618567, 3870909, 24441021, 155510523, 996109245, 6418243575, 41572149615, 270536350545, 1767990955980, 11598120859860, 76347126498420, 504148079084940, 3338585176489560, 22166530404950520, 147525638070221640, 983978335278966456, 6576191509703182677, 44031626057441376423
Offset: 1

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Radius of convergence is r = 1/7, where r = r^2/(1-6*r), with A(r) = 1.
Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

Examples

			G.f.: A(x) = x + 3*x^2 + 15*x^3 + 81*x^4 + 462*x^5 + 2718*x^6 + 16344*x^7 + 99900*x^8 + 618567*x^9 + 3870909*x^10 + 24441021*x^11 + 155510523*x^12 +...
where A(x)^2 = A(x^2/(1-6*x)).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 252*x^5 + 1635*x^6 + 10638*x^7 + 69417*x^8 + 454248*x^9 + 2980614*x^10 + 19609380*x^11 + 129337686*x^12 +...
A( x/(1+3*x) ) = x + 6*x^3 + 57*x^5 + 630*x^7 + 7584*x^9 + 96552*x^11 + 1277937*x^13 + 17393454*x^15 + 241666275*x^17 + 3410638362*x^19 + 48723929721*x^21 +...
A( x^2/(1-9*x^2) ) = x^2 + 12*x^4 + 150*x^6 + 1944*x^8 + 25977*x^10 + 355932*x^12 + 4975974*x^14 + 70684920*x^16 + 1016911392*x^18 + 14778827136*x^20 +...
where A( x^2/(1-9*x^2) ) = A( x/(1+3*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 3*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 +...+ A264413(n)*x^(2*n) +...
such that B(x) = F(x^2) + 3*x = F(x)^2 - 9*x and F(x) is the g.f. of A264413.

Links

Crossrefs

Cf. A264413, A264233, A264224, A264226, A264227.

Programs

  • Mathematica
    max = 25; For[A = x; i = 1, i <= max, i++, A = Sqrt[Normal[A] /. x -> x^2/(1 - 6*x + x*O[x]^max)]]; CoefficientList[A, x] // Rest (* Jean-François Alcover, Nov 22 2016 *)
  • PARI
    {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-6*x +x*O(x^n))) ) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. also satisfies:
(1) A(x) = -A( -x/(1-6*x) ).
(2) A( x/(1+3*x) ) = -A( -x/(1-3*x) ), an odd function.
(3) A( x/(1+3*x) )^2 = A( x^2/(1-9*x^2) ), an even function.
(4) A(x)^4 = A( x^4/((1-6*x)*(1-6*x-6*x^2)) ).
(5) [x^(2*n+1)] (x/A(x))^(2*n) = 0 for n>=0.
(6) [x^(2^n+k)] (x/A(x))^(2^n) = 0 for k=1..2^n-1, n>=1.
Given g.f. A(x), let F(x) denote the g.f. of A264413, then:
(7) A(x) = F(A(x))^2 * x/(1+9*x),
(8) A(x) = F(A(x)^2) * x/(1-3*x),
(9) A( x/(F(x)^2 - 9*x) ) = x,
(10) A( x/(F(x^2) + 3*x) ) = x,
where F(x)^2 = F(x^2) + 12*x.
Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-6)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.

A264413 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 12*x.

Original entry on oeis.org

1, 6, -15, 90, -660, 5310, -45765, 413640, -3864345, 37014120, -361577790, 3588484140, -36079979085, 366728363460, -3762120325140, 38901621985290, -405039437707575, 4242802537386450, -44681704461745740, 472795814216587140, -5024232597805717410, 53596341229925979360, -573736849659978481665, 6161218734911098973490, -66355728143871653462745
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2015

Keywords

Examples

			G.f.: A(x) = 1 + 6*x - 15*x^2 + 90*x^3 - 660*x^4 + 5310*x^5 - 45765*x^6 + 413640*x^7 - 3864345*x^8 + 37014120*x^9 - 361577790*x^10 +...
where
A(x)^2 = 1 + 12*x + 6*x^2 - 15*x^4 + 90*x^6 - 660*x^8 + 5310*x^10 - 45765*x^12 + 413640*x^14 - 3864345*x^16 + 37014120*x^18 - 361577790*x^20 +...
so that A(x)^2 = A(x^2) + 12*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 3*x) ), then
G(x) = x + 3*x^2 + 15*x^3 + 81*x^4 + 462*x^5 + 2718*x^6 + 16344*x^7 + 99900*x^8 + 618567*x^9 + 3870909*x^10 +...+ A264225(n)*x^n +...
such that G(x)^2 = G( x^2/(1-6*x) ).

Links

Crossrefs

Cf. A271935, A264225, A264233, A264412, A264414, A264415.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 12*x +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

Given g.f. A(x), let G(x) denote the g.f. of A264225, then:
(1) G( x/(A(x)^2 - 9*x) ) = x,
(2) G( x/(A(x^2) + 3*x) ) = x,
(3) A(G(x))^2 = (1+9*x) * G(x)/x,
(4) A(G(x)^2) = (1-3*x) * G(x)/x,
where G(x)^2 = G( x^2/(1-6*x) ).

A264232 G.f. satisfies: A(x)^2 = A( x^2/(1-6*x)^2 ).

Original entry on oeis.org

1, 6, 39, 270, 1959, 14706, 113166, 887004, 7050837, 56672622, 459646488, 3756181248, 30893173038, 255509028612, 2123685458190, 17728918028172, 148590381782418, 1249839423702828, 10547139497197887, 89271390230559918, 757673193636234279, 6446893091203601298, 54983813851196942292, 469959567684908644440
Offset: 1

Views

Author

Paul D. Hanna, Nov 16 2015

Keywords

Comments

Radius of convergence is r = 1/9 where r = r^2/(1-6*r)^2 with A(r) = 1.
Compare to: C(x)^2 = C( x^2/(1-2*x)^2 ) where C(x) = (1-2*x-sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers A000108.

Examples

			G.f.: A(x) = x + 6*x^2 + 39*x^3 + 270*x^4 + 1959*x^5 + 14706*x^6 + 113166*x^7 + 887004*x^8 + 7050837*x^9 + 56672622*x^10 + 459646488*x^11 + 3756181248*x^12 +...
where A( x^2/(1-6*x)^2 ) = A(x)^2,
A( x^2/(1-6*x)^2 ) = x^2 + 12*x^3 + 114*x^4 + 1008*x^5 + 8679*x^6 + 73980*x^7 + 628506*x^8 + 5336928*x^9 + 45351591*x^10 + 385869348*x^11 + 3287962710*x^12 +...
Also, A( x/(1+6*x) ) = A(x^2)^(1/2),
A( x/(1+6*x) ) = x + 3*x^3 + 15*x^5 + 90*x^7 + 597*x^9 + 4212*x^11 + 30942*x^13 + 233766*x^15 + 1802706*x^17 + 14122359*x^19 + 112033791*x^21 + 898024320*x^23 +...
Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
B(x) = 1 + 6*x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 + 990198*x^22 +...+ A264412(n)*x^(2*n) +...
such that B(x) = F(x^2) + 6*x = F(x)^2 where F(x) is the g.f. of A264412.

Links

Crossrefs

Cf. A264412, A264224, A264233, A260650.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,#binary(n+1), A = sqrt( subst(A,x, x^2/(1-6*x +x*O(x^n))^2) ) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x) = -A( -x/(1-12*x) ).
(2) A(x^2) = A( x/(1+6*x) )^2 = A( -x/(1-6*x) )^2.
(3) A( x/(1+3*x)^2 ) = -A( -x/(1-3*x)^2 ), an odd function.
(4) A( x/(1+3*x)^2 )^2 = A( x^2/(1+9*x^2)^2 ), an even function.
(5) A( x/(1+4*x) ) = G(x) = Sum_{n>=1} A264224(n)*x^n where G(x)^2 = G( x^2/(1-4*x) ).
(6) A( x/(1+8*x) ) = -G(-x) = Sum_{n>=1} (-1)^(n-1) * A264224(n)*x^n where G(x)^2 = G( x^2/(1-4*x) ).
Sum_{k=0..n} binomial(n,k) * (-6)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = A264224(n+1) for n>=0.
Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * A264224(n+1) for n>=0.
Showing 1-3 of 3 results.

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