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.

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A264412 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 6*x.

Original entry on oeis.org

1, 3, -3, 9, -33, 126, -513, 2214, -9876, 45045, -209493, 990198, -4741191, 22946247, -112079214, 551793303, -2735330190, 13641353118, -68394016548, 344539469889, -1743035351958, 8851923849123, -45110440515753, 230615809867476, -1182376529280117, 6078184963674498, -31322206517658453, 161774639164275552, -837290923919381322
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2015

Keywords

Examples

			G.f.: A(x) = 1 + 3*x - 3*x^2 + 9*x^3 - 33*x^4 + 126*x^5 - 513*x^6 + 2214*x^7 - 9876*x^8 + 45045*x^9 +...
where
A(x)^2 = 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 +...
so that A(x)^2 = A(x^2) + 6*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 2*x) ), then
G(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 103*x^5 + 422*x^6 + 1774*x^7 + 7604*x^8 + 33109*x^9 + 146042*x^10 +...+ A264224(n)*x^n +...
such that G(x)^2 = G( x^2/(1-4*x) ) and A(G(x))^2 = (1+4*x) * G(x)/x.

Links

Crossrefs

Cf. A271930, A264224, A264413, A264414, A264415.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 6*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 A264224, then:
(1) G( x/(A(x)^2 - 4*x) ) = x,
(2) G( x/(A(x^2) + 2*x) ) = x,
(3) A(G(x))^2 = (1+4*x) * G(x)/x,
(4) A(G(x)^2) = (1-2*x) * G(x)/x,
where G(x)^2 = G( x^2/(1-4*x) ).
a(n) ~ c * (-1)^(n+1) * d^n / n^(3/2), where d = 5.46806882358680646837..., c = 0.268849330049069376... . - Vaclav Kotesovec, Nov 18 2015

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.

A274484 G.f. satisfies: A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).

Original entry on oeis.org

1, 2, 6, 20, 71, 262, 994, 3852, 15183, 60686, 245410, 1002300, 4128448, 17129920, 71529800, 300355184, 1267386163, 5371101382, 22850230642, 97546995260, 417717017392, 1793765580704, 7722405668232, 33323153856880, 144099312039391, 624347587536782, 2710036186345914, 11782865084403212, 51310167663855675, 223762749750806942, 977155903597684074, 4272633455348970588, 18704696346822470087, 81978422471165944654
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2016

Keywords

Comments

Radius of convergence of g.f. A(x) is r = (5 - sqrt(17))/4 where r = r^2/(1-4*r+2*r^2) with A(r) = 1.
Compare g.f. with the identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) ).

Examples

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 71*x^5 + 262*x^6 + 994*x^7 + 3852*x^8 + 15183*x^9 + 60686*x^10 + 245410*x^11 + 1002300*x^12 +...
such that A( x^2/(1-4*x+2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 258*x^6 + 1048*x^7 + 4288*x^8 + 17664*x^9 + 73223*x^10 + 305292*x^11 + 1279632*x^12  + 5389632*x^13 + 22800926*x^14 +...
The g.f. of A260650, F(x), begins:
A( x/(1 - 2*x) ) = x + 4*x^2 + 18*x^3 + 88*x^4 + 455*x^5 + 2444*x^6 + 13486*x^7 + 75912*x^8 + 433935*x^9 + 2511388*x^10 +...
and satisfies: F(x)^2 = F( x^2/(1 - 4*x)^2 ).
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - 2*x^2 + 2*x^3 - 3*x^5 + 4*x^6 - 2*x^7 + 2*x^9 - 10*x^10 + 18*x^11 - 39*x^13 + 28*x^14 + 40*x^15 - 142*x^17 - 84*x^18 + 620*x^19 - 1735*x^21 + 260*x^22 + 4532*x^23 +...
which is related to A107087 by:
x/Series_Reversion(A(x)) = 1 + 2*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 - 233*x^16 + 668*x^18 - 1949*x^20 +...+ A107087(n)*x^(2*n) +...
The g.f. G(x) of A107087 begins:
G(x) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 + 668*x^9 - 1949*x^10 + 5802*x^11 - 17503*x^12 +...
where G(x)^2 = G(x^2) + 4*x.
Also, we have A(x/(1 + 2*x + 3*x^2))^2 = A(x^2/(1 + 4*x^2 + 9*x^4)), where the series begin:
A(x/(1 + 2*x + 3*x^2)) = x - x^3 - 2*x^5 + 6*x^7 - x^9 - 3*x^11 - 30*x^13 - 66*x^15 + 715*x^17 - 747*x^19 - 4028*x^21 + 9424*x^23 + 8790*x^25 +...
A(x^2/(1 + 4*x^2 + 9*x^4)) = x^2 - 2*x^4 - 3*x^6 + 16*x^8 - 10*x^10 - 28*x^12 - 14*x^14 - 72*x^16 + 1647*x^18 - 3014*x^20 - 10145*x^22 + 38784*x^24 +...
which is equal to A(x/(1 + 2*x + 3*x^2))^2.

Links

Crossrefs

Cf. A107087, A260650, A264224, A274483, A274478, A274479.
Cf. A264225, A357547, A357548.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-4*x+2*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = -A( -x/(1 - 4*x) ). - Paul D. Hanna, Nov 30 2022
(2) A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).
(3) A( x/(1 + 2*x + 3*x^2) )^2 = A( x^2/(1 + 4*x^2 + 9*x^4) ).
(4) A( x/(1 + 2*x) )^2 = x * A( x/(1 - 2*x) ).
(5) A( x/(1 - 2*x) )^2 = A( x^2/(1 - 8*x + 14*x^2) ).
Let G(x) denote the g.f. of A107087, where G(x)^2 = G(x^2) + 4*x, then g.f. A(x) satisfies:
(6) A(x) = x/(1-2*x) * G( A(x)^2 ),
(7) A(x) = Series_Reversion( x/(G(x)^2 - 2*x) ),
(8) G(x) = sqrt( x/Series_Reversion(A(x)) + 2*x ),
(9) G(x^2) = x/Series_Reversion(A(x)) - 2*x,
(10) A( x/(G(x)^2 - 2*x) ) = x,
(11) A( x/(G(x^2) + 2*x) ) = x,
(12) A(x)^2/(G(A(x)^4) + 2*A(x)^2) = x^2/(1 - 4*x + 2*x^2).

A264226 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-8*x) ), with A(0) = 0.

Original entry on oeis.org

1, 4, 26, 184, 1371, 10524, 82446, 655624, 5274581, 42835444, 350607226, 2888950904, 23943016426, 199450842504, 1669044107916, 14024053212624, 118272485941116, 1000814156934384, 8494876225031496, 72307674880328544, 617074982874821901, 5278745007753158724, 45256869801034564986, 388802380782229815384, 3346570416790776555756
Offset: 1

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Radius of convergence is r = 1/9, where r = r^2/(1-8*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 + 4*x^2 + 26*x^3 + 184*x^4 + 1371*x^5 + 10524*x^6 + 82446*x^7 + 655624*x^8 + 5274581*x^9 + 42835444*x^10 + 350607226*x^11 +...
where A(x)^2 = A(x^2/(1-8*x)).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 576*x^5 + 4890*x^6 + 41584*x^7 + 354232*x^8 + 3022592*x^9 + 25833819*x^10 + 221156920*x^11 + 1896267356*x^12 +...
(A(x)/x)^(1/2) = 1 + 2*x + 11*x^2 + 70*x^3 + 485*x^4 + 3522*x^5 + 26394*x^6 + 202332*x^7 + 1578140*x^8 + 12480040*x^9 + 99817421*x^10 + 805999682*x^11 +...
(A(x)/x)^(1/4) = 1 + x + 5*x^2 + 30*x^3 + 200*x^4 + 1411*x^5 + 10336*x^6 + 77775*x^7 + 597285*x^8 + 4661580*x^9 + 36864795*x^10 + 294769500*x^11 +...
A( x/(1+4*x) ) = x + 10*x^3 + 155*x^5 + 2750*x^7 + 52565*x^9 + 1055850*x^11 + 21979050*x^13 + 469891500*x^15 + 10252631420*x^17 + 227274091400*x^19 +...
A( x^2/(1-16*x^2) ) = x^2 + 20*x^4 + 410*x^6 + 8600*x^8 + 184155*x^10 + 4015500*x^12 + 88932750*x^14 + 1995785000*x^16 + 45286852565*x^18 +...
where A( x^2/(1-16*x^2) ) = A( x/(1+4*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 4*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 + 16953750*x^14 - 268652880*x^16 + 4365638550*x^18 +...+ A264414(n)*x^(2*n) +...
such that B(x) = F(x^2) + 4*x = F(x)^2 - 16*x and F(x) is the g.f. of A264414.

Links

Crossrefs

Cf. A264414, A264224, A264225, A264227.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-8*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/(1-8*x) ).
(2) A( x/(1+4*x) ) = -A( -x/(1-4*x) ), an odd function.
(3) A( x/(1+4*x) )^2 = A( x^2/(1-16*x^2) ), an even function.
(4) A(x)^4 = A( x^4/((1-8*x)*(1-8*x-8*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 A264414, then:
(7) A(x) = F(A(x))^2 * x/(1+16*x),
(8) A(x) = F(A(x)^2) * x/(1-4*x),
(9) A( x/(F(x)^2 - 16*x) ) = x,
(10) A( x/(F(x^2) + 4*x) ) = x,
where F(x)^2 = F(x^2) + 20*x.
Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.

A264227 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-10*x) ), with A(0) = 0.

Original entry on oeis.org

1, 5, 40, 350, 3220, 30500, 294625, 2886875, 28598035, 285786575, 2876602225, 29131678625, 296574083425, 3033183585125, 31148390740375, 321040368434375, 3319845741478030, 34433523106882550, 358129419509956150, 3734203057793066750, 39027568927659117700, 408777143934160983500, 4290195975642644398000, 45111124579414224095000
Offset: 1

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Radius of convergence is r = 1/11, where r = r^2/(1-10*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 + 5*x^2 + 40*x^3 + 350*x^4 + 3220*x^5 + 30500*x^6 + 294625*x^7 + 2886875*x^8 + 28598035*x^9 + 285786575*x^10 + 2876602225*x^11 +...
where A(x)^2 = A(x^2/(1-10*x)).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 105*x^4 + 1100*x^5 + 11540*x^6 + 121200*x^7 + 1274350*x^8 + 13414000*x^9 + 141353220*x^10 + 1491161000*x^11 + 15747360500*x^12 +...
A( x/(1+5*x) ) = x + 15*x^3 + 345*x^5 + 9000*x^7 + 251160*x^9 + 7328475*x^11 + 220880925*x^13 + 6824229750*x^15 + 214969962405*x^17 + 6877343600775*x^19 +...
A( x^2/(1-25*x^2) ) = x^2 + 30*x^4 + 915*x^6 + 28350*x^8 + 891345*x^10 + 28401750*x^12 + 915916500*x^14 + 29852415000*x^16 + 982068551160*x^18 +...
where A( x^2/(1-25*x^2) ) = A( x/(1+5*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 5*x + 15*x^2 - 105*x^4 + 1575*x^6 - 29190*x^8 + 603225*x^10 - 13352850*x^12 + 309605625*x^14 - 7422255645*x^16 +...+ A264415(n)*x^(2*n) +...
such that B(x) = F(x^2) + 5*x = F(x)^2 - 25*x and F(x) is the g.f. of A264415.

Links

Crossrefs

Cf. A264415, A264224, A264225, A264226.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-10*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/(1-10*x) ).
(2) A( x/(1+5*x) ) = -A( -x/(1-5*x) ), an odd function.
(3) A( x/(1+5*x) )^2 = A( x^2/(1-25*x^2) ), an even function.
(4) A(x)^4 = A( x^4/((1-10*x)*(1-10*x-10*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 A264415, then:
(7) A(x) = F(A(x))^2 * x/(1+25*x),
(8) A(x) = F(A(x)^2) * x/(1-5*x),
(9) A( x/(F(x)^2 - 25*x) ) = x,
(10) A( x/(F(x^2) + 5*x) ) = x,
where F(x)^2 = F(x^2) + 30*x.
Sum_{k=0..n} binomial(n,k) * (-5)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) *(-10)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.

A260650 G.f. satisfies: A(x)^2 = A( x^2/(1-4*x)^2 ).

Original entry on oeis.org

1, 4, 18, 88, 455, 2444, 13486, 75912, 433935, 2511388, 14684422, 86611848, 514704064, 3078845696, 18523994024, 112026315616, 680626958899, 4152411174284, 25428402204982, 156247439709832, 963048223399984, 5952595420121536, 36887847899094888, 229132114803540320, 1426367728966653535, 8897049258366111004
Offset: 1

Views

Author

Paul D. Hanna, Nov 16 2015

Keywords

Comments

Radius of convergence is r = (9 - sqrt(17))/32 where r = r^2/(1-4*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 + 4*x^2 + 18*x^3 + 88*x^4 + 455*x^5 + 2444*x^6 + 13486*x^7 + 75912*x^8 + 433935*x^9 + 2511388*x^10 + 14684422*x^11 + 86611848*x^12 +...
where A( x^2/(1-4*x)^2 ) = A(x)^2,
A( x^2/(1-4*x)^2 ) = x^2 + 8*x^3 + 52*x^4 + 320*x^5 + 1938*x^6 + 11696*x^7 + 70648*x^8 + 427776*x^9 + 2597831*x^10 + 15824664*x^11 + 96687516*x^12 +...
Also, A( x/(1+4*x) ) = A(x^2)^(1/2),
A( x/(1+4*x) ) = x + 2*x^3 + 7*x^5 + 30*x^7 + 143*x^9 + 726*x^11 + 3840*x^13 + 20904*x^15 + 116275*x^17 + 657798*x^19 + 3772912*x^21 + 21890152*x^23 +...
Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
B(x) = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 - 233*x^16 + 668*x^18 - 1949*x^20 + 5802*x^22 +...+ A107087(n)*x^(2*n) +...
such that B(x) = F(x^2) + 4*x = F(x)^2 where F(x) is the g.f. of A107087.

Links

Crossrefs

Cf. A264232, A264224, A107087, A107088.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,#binary(n+1), A = sqrt( subst(A,x, x^2/(1-4*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/(1-8*x) ).
(2) A(x^2) = A( x/(1+4*x) )^2 = A( -x/(1-4*x) )^2.
(3) A( x/(1+2*x)^2 ) = -A( -x/(1-2*x)^2 ), an odd function.
(4) A( x/(1+2*x)^2 )^2 = A( x^2/(1+4*x^2)^2 ), an even function.

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.

A357547 a(n) = coefficient of x^n in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).

Original entry on oeis.org

1, 2, 9, 38, 176, 832, 4039, 19938, 99861, 506042, 2590099, 13370898, 69540016, 364028992, 1916585714, 10142059868, 53911982971, 287736310102, 1541243386819, 8282387269058, 44638363790176, 241216694913632, 1306608966475854, 7092980525443588, 38581011402034156
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2022

Keywords

Comments

Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*r^2), with A(r) = 1.
Related identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) );
here, a = 2, b = 6.

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 38*x^4 + 176*x^5 + 832*x^6 + 4039*x^7 + 19938*x^8 + 99861*x^9 + 506042*x^10 + 2590099*x^11 + 13370898*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 112*x^5 + 585*x^6 + 3052*x^7 + 16018*x^8 + 84384*x^9 + 446384*x^10 + 2370240*x^11 + 12631104*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 4*x^3 + 15*x^4 + 65*x^5 + 291*x^6 + 1356*x^7 + 6474*x^8 + 31555*x^9 + 156315*x^10 + 784924*x^11 + ... + A357785(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 5*x^2 - 10*x^4 + 50*x^6 - 305*x^8 + 2025*x^10 - 14400*x^12 + 107500*x^14 - 829415*x^16 + 6559700*x^18 - 52908950*x^20 + ...

Links

Crossrefs

Cf. A357785, A264224, A274483, A274484, A357548.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 4*x^2 +x*O(x^n)) ) )
); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x/(1 + 2*x + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3.a) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
(3.b) A(x)^2 = -A( -x^2/(1 - 4*x - 8*x^2) ).
(4.a) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ).
(4.b) A( x/(1 + 2*x) )^2 = -A( -x^2/(1 - 12*x^2) ).
(4.c) A( x/(1 + 2*x) )^2 = A( -x/(1 - 2*x) )^2.

A357548 a(n) = coefficient of x^n in A(x) where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).

Original entry on oeis.org

1, 2, 11, 50, 261, 1362, 7344, 40112, 222338, 1245476, 7043605, 40153390, 230518723, 1331576430, 7733934030, 45138530004, 264596552838, 1557101158092, 9195520745412, 54477134410680, 323668083179382, 1928047124332764, 11512382184408072, 68889282756213840
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2022

Keywords

Comments

Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*r^2), with A(r) = 1.
Related identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) );
here, a = 2, b = 8.

Examples

			G.f.: A(x) = x + 2*x^2 + 11*x^3 + 50*x^4 + 261*x^5 + 1362*x^6 + 7344*x^7 + 40112*x^8 + 222338*x^9 + 1245476*x^10 + 7043605*x^11 + 40153390*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 144*x^5 + 843*x^6 + 4868*x^7 + 28378*x^8 + 165664*x^9 + 971013*x^10 + 5708132*x^11 + 33660362*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + ... + A357786(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 7*x^2 - 21*x^4 + 147*x^6 - 1260*x^8 + 11907*x^10 - 120540*x^12 + 1279047*x^14 - 14029428*x^16 + 157788183*x^18 + ...

Links

Crossrefs

Cf. A357786, A264224, A274483, A274484, A357547.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 8*x^2 +x*O(x^n)) ) )
); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ).
(5) A( x/(1 + 4*x) )^2 = A( x^2/(1 + 4*x - 8*x^2) ).

A274483 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1 - 4*x - 2*x^2) ).

Original entry on oeis.org

1, 2, 8, 32, 138, 612, 2784, 12896, 60635, 288614, 1388104, 6735808, 32938438, 162156828, 803026176, 3997462368, 19991321445, 100387500906, 505950179016, 2558352514272, 12974595610122, 65975538192036, 336293496474144, 1717927441213152, 8793426613714734, 45092543870052092, 231621905868337424, 1191586088094887936, 6138909938284313524, 31668826322371245256, 163571372589617459584, 845826517521629901888, 4378463647900723645800
Offset: 1

Views

Author

Paul D. Hanna, Aug 03 2016

Keywords

Comments

Radius of convergence is r = (sqrt(33) - 5)/4 where A(r) = 1.
Compare the g.f. with the identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) ).

Examples

			G.f.: A(x) = x + 2*x^2 + 8*x^3 + 32*x^4 + 138*x^5 + 612*x^6 + 2784*x^7 + 12896*x^8 + 60635*x^9 + 288614*x^10 + 1388104*x^11 + 6735808*x^12 +...
such that A( x^2/(1 - 4*x - 2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 20*x^4 + 96*x^5 + 468*x^6 + 2288*x^7 + 11248*x^8 + 55552*x^9 + 275610*x^10 + 1373192*x^11 + 6869096*x^12 +...
The series reversion of g.f. A(x) begins
Series_Reversion(A(x)) = x - 2*x^2 + 8*x^4 - 10*x^5 - 24*x^6 + 64*x^7 + 64*x^8 - 327*x^9 - 172*x^10 + 1664*x^11 + 480*x^12 - 8858*x^13 - 1328*x^14 + 49344*x^15 + 3584*x^16 - 286432*x^17 - 9714*x^18 + 1723264*x^19 + 26800*x^20 - 10669788*x^21 - 73768*x^22 + 67557440*x^23 + 200448*x^24 +...
Now compare the expansion given by
x/Series_Reversion(A(x)) = 1 + 2*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 + 20184*x^14 - 122883*x^16 + 766464*x^18 - 4875378*x^20 + 31507728*x^22 - 206278686*x^24 + 1365201252*x^26 - 9118841784*x^28 + 61393574760*x^30 - 416193047280*x^32 + 2838492444204*x^34 +...
to the series G(x) such that G(x)^2 = G(x^2) + 8*x, which begins
G(x) = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 - 122883*x^8 + 766464*x^9 - 4875378*x^10 + 31507728*x^11 - 206278686*x^12 +...
and equals the square of the g.f. of A228711.

Crossrefs

Cf. A228711, A274484, A264224.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-4*x-2*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A( x/(1 + 2*x + 5*x^2) )^2 = A( x^2/(1 + 4*x^2 + 25*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A( x/(1 + 2*x) ) = -A( -x/(1 - 2*x) ), an odd function.
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 6*x^2) ), an even function.
Given G(x) such that G(x)^2 = G(x^2) + 8*x, then g.f. A(x) satisfies:
(5) A(x) = x/(1-2*x) * G( A(x)^2 ),
(6) A(x) = Series_Reversion( x/(G(x)^2 - 6*x) ),
(7) G(x) = sqrt( x/Series_Reversion(A(x)) + 6*x ),
(8) G(x^2) = x/Series_Reversion(A(x)) - 2*x,
(9) A( x/(G(x)^2 - 6*x) ) = x,
(10) A( x/(G(x^2) + 2*x) ) = x,
(11) A(x)^2/(G(A(x)^4) + 2*A(x)^2) = x^2/(1 - 4*x - 2*x^2).
Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.
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