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 1-6 of 6 results.

A264224 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-4*x) ), with A(0) = 0.

Original entry on oeis.org

1, 2, 7, 26, 103, 422, 1774, 7604, 33109, 146042, 651256, 2931392, 13301038, 60775340, 279393742, 1291311620, 5996491666, 27962898020, 130883946751, 614664907706, 2895279687655, 13674609742598, 64744203198388, 307221794213768, 1460778188820220, 6958635514922552, 33205258829750809, 158699556581760134
Offset: 1

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Radius of convergence is r = 1/5, where r = r^2/(1-4*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 + 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 + 651256*x^11 + 2931392*x^12 +...
where A(x)^2 = A(x^2/(1-4*x)).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 18*x^4 + 80*x^5 + 359*x^6 + 1620*x^7 + 7354*x^8 + 33568*x^9 + 154023*x^10 + 710156*x^11 + 3289142*x^12 + 15297744*x^13 +...
sqrt(A(x)/x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 144*x^5 + 582*x^6 + 2418*x^7 + 10266*x^8 + 44353*x^9 + 194395*x^10 +...+ A264231(n)*x^n +...
A( x/(1+2*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 +...
A( x^2/(1-4*x^2) ) = x^2 + 6*x^4 + 39*x^6 + 270*x^8 + 1959*x^10 + 14706*x^12 + 113166*x^14 + 887004*x^16 + 7050837*x^18 + 56672622*x^20 + 459646488*x^22 +...
where A( x^2/(1-4*x^2) ) = A( x/(1+2*x) )^2.
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 2*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 +...+ A264412(n)*x^(2*n) +...
such that B(x) = F(x^2) + 2*x = F(x)^2 - 4*x and F(x) is the g.f. of A264412.
PARTICULAR VALUES.
A(1/5) = 1.
A(-1/5) = -A(1/9) = -0.15262256991492310976978497600904...
A(1/6)^2 = A(1/12) = 0.10315964246752710052686298695713...
A(1/6)^4 = A(1/96) = 0.01064191183402802084987998396215...
A(1/7)^2 = A(1/21) = 0.053075120978549663441827849989065...
A(1/7)^4 = A(1/357) = 0.002816968466887682583828696137137...
A(1/8)^2 = A(1/32) = 0.033445065874191867268119916059631...
A(1/8)^4 = A(1/896) = 0.001118572431329033410718706838540...
A(1/9)^2 = A(1/45) = 0.0232936488474355927381514600230212...

Links

Crossrefs

Cf. A264231, A264232, A264412, A264225, A264226, A264227.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x,x^2/(1-4*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/(1-4*x) ).
(2) A( x/(1+2*x) ) = -A( -x/(1-2*x) ), an odd function.
(3) A( x/(1+2*x) )^2 = A( x^2/(1-4*x^2) ), an even function.
(4) A(x)^4 = A( x^4/((1-4*x)*(1-4*x-4*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 A264412, then:
(7) A(x) = F(A(x))^2 * x/(1+4*x),
(8) A(x) = F(A(x)^2) * x/(1-2*x),
(9) A( x/(F(x)^2 - 4*x) ) = x,
(10) A( x/(F(x^2) + 2*x) ) = x,
where F(x)^2 = F(x^2) + 6*x.
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.
Sum_{k=0..n} binomial(n,k) * (+4)^(n-k) * a(k+1) = A264232(n+1) for n>=0.
Sum_{k=0..n} binomial(n,k) * (-8)^(n-k) * a(k+1) = (-1)^n * A264232(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) ).

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.

A264233 G.f. satisfies: A(x)^2 = A( x^2/(1-12*x)^2 ).

Original entry on oeis.org

1, 12, 150, 1944, 25977, 355932, 4975974, 70684920, 1016911392, 14778827136, 216547264296, 3194332332192, 47384274750705, 706221689838300, 10568432343600990, 158713925474269080, 2390963478663939555, 36119150645827725540, 547001314170524048970, 8302813348383238118760, 126288497159001902128185, 1924561894757711270308380
Offset: 1

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Author

Paul D. Hanna, Nov 17 2015

Keywords

Comments

Radius of convergence is r = 1/16 where r = r^2/(1-12*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 + 12*x^2 + 150*x^3 + 1944*x^4 + 25977*x^5 + 355932*x^6 + 4975974*x^7 + 70684920*x^8 + 1016911392*x^9 + 14778827136*x^10 + 216547264296*x^11 +...
where A( x^2/(1-12*x)^2 ) = A(x)^2,
A( x^2/(1-12*x)^2 ) = x^2 + 24*x^3 + 444*x^4 + 7488*x^5 + 121110*x^6 + 1918512*x^7 + 30066552*x^8 + 468571392*x^9 + 7281721209*x^10 + 113007681720*x^11 +...
Also, A( x/(1+12*x) ) = A(x^2)^(1/2),
A( x/(1+12*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 +...
Let B(x) = x/Series_Reversion( A(x) ), so that A(x) = x*B(A(x)), then
B(x) = 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 +...+ A264413(n)*x^(2*n) +...
such that B(x) = F(x^2) + 12*x = F(x)^2 where F(x) is the g.f. of A264413.

Crossrefs

Cf. A264413, A264225, A264232.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,#binary(n+1), A = sqrt( subst(A,x, x^2/(1-12*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/(1-24*x) ).
(2) A(x^2) = A( x/(1+12*x) )^2 = A( -x/(1-12*x) )^2.
(3) A( x/(1+6*x)^2 ) = -A( -x/(1-6*x)^2 ), an odd function.
(4) A( x/(1+6*x)^2 )^2 = A( x^2/(1+36*x^2)^2 ), an even function.
(5) A( x/(1+9*x) ) = G(x) = Sum_{n>=1} A264225(n)*x^n where G(x)^2 = G( x^2/(1-6*x) ).
(6) A( x/(1+15*x) ) = -G(-x) = Sum_{n>=1} (-1)^(n-1) * A264225(n)*x^n where G(x)^2 = G( x^2/(1-6*x) ).
Sum_{k=0..n} binomial(n,k) *(-12)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-9)^(n-k) * a(k+1) = A264225(n+1) for n>=0.
Sum_{k=0..n} binomial(n,k) *(-15)^(n-k) * a(k+1) = (-1)^n * A264225(n+1) for n>=0.
Showing 1-6 of 6 results.

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