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 1-5 of 5 results.

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

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) ).

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.

A264415 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 30*x.

Original entry on oeis.org

1, 15, -105, 1575, -29190, 603225, -13352850, 309605625, -7422255645, 182481301800, -4575894819300, 116581172754375, -3009161401332975, 78523515330379875, -2068113764887828875, 54904020923799337500, -1467692309121298737960, 39472725372798507822900, -1067296235915278105855650, 28996357915496677935088125, -791147023483262777604486675, 21669197341488265510394307750
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2015

Keywords

Examples

			G.f.: A(x) = 1 + 15*x - 105*x^2 + 1575*x^3 - 29190*x^4 + 603225*x^5 - 13352850*x^6 + 309605625*x^7 +...
where
A(x)^2 = 1 + 30*x + 15*x^2 - 105*x^4 + 1575*x^6 - 29190*x^8 + 603225*x^10 - 13352850*x^12 + 309605625*x^14 +...
so that A(x)^2 = A(x^2) + 30*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 5*x) ), then
G(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 +...+ A264227(n)*x^n +...
such that G(x)^2 = G( x^2/(1-10*x) ).

Links

Crossrefs

Cf. A264227, A264412, A264413, A264414.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 30*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 A264227, then:
(1) G( x/(A(x)^2 - 25*x) ) = x,
(2) G( x/(A(x^2) + 5*x) ) = x,
(3) A(G(x))^2 = (1+25*x) * G(x)/x,
(4) A(G(x)^2) = (1-5*x) * G(x)/x,
where G(x)^2 = G( x^2/(1-10*x) ).

A271957 G.f. A(x) satisfies: A(x) = A( x^2 + 10*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 5, 40, 375, 3845, 41825, 474450, 5552250, 66548785, 812875800, 10082125950, 126637168125, 1607562407775, 20591392666250, 265810034489750, 3454516382881875, 45162288467005155, 593528625987396725, 7836767285955169200, 103908861022437312375, 1382961699685548183750, 18469547560714428659250, 247433242662040209056250, 3324296142183357299203125, 44779542961314348791789400, 604655933814703316140014375
Offset: 1

Views

Author

Paul D. Hanna, Apr 17 2016

Keywords

Comments

Compare the g.f. to the following related identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
(2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).

Examples

			G..f.: A(x) = x + 5*x^2 + 40*x^3 + 375*x^4 + 3845*x^5 + 41825*x^6 + 474450*x^7 + 5552250*x^8 + 66548785*x^9 + 812875800*x^10 + 10082125950*x^11 + 126637168125*x^12 +...
where A(x)^2 = A( x^2 + 10*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 105*x^4 + 1150*x^5 + 13040*x^6 + 152100*x^7 + 1815375*x^8 + 22078750*x^9 + 272728845*x^10 + 3412891200*x^11 + 43178951325*x^12 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 5*x^2 + 10*x^3 - 45*x^5 + 450*x^7 - 5535*x^9 + 75600*x^11 - 1106100*x^13 + 16953750*x^15 +...+ A264414(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 5*x^2 where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.

Links

Crossrefs

Cf. A264414, A271930, A271935, A271931, A271934.

Programs

  • PARI
    {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 10*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies: A( x*G(x^2) - 5*x^2 ) = x, where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.
Showing 1-5 of 5 results.

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