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.

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) ).

A264414 G.f. A(x) satisfies: A(x)^2 = A(x^2) + 20*x.

Original entry on oeis.org

1, 10, -45, 450, -5535, 75600, -1106100, 16953750, -268652880, 4365638550, -72354858300, 1218356280000, -20784495119850, 358457180010750, -6239532583193625, 109476057598087500, -1934128026918961515, 34378012275668994150, -614328464414815220025, 11030366153872043358750, -198899407327466712808800, 3600377821710426377668500
Offset: 0

Views

Author

Paul D. Hanna, Nov 12 2015

Keywords

Examples

			G.f.: A(x) = 1 + 10*x - 45*x^2 + 450*x^3 - 5535*x^4 + 75600*x^5 - 1106100*x^6 +...
where
A(x)^2 = 1 + 20*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 +...
so that A(x)^2 = A(x^2) + 20*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 4*x) ), then
G(x) = x + 4*x^2 + 28*x^3 + 208*x^4 + 1702*x^5 + 14584*x^6 + 129808*x^7 + 1187008*x^8 + 11089153*x^9 + 105370660*x^10 +...+ A264226(n)*x^n +...
such that G(x)^2 = G( x^2/(1-8*x) ).

Links

Crossrefs

Cf. A271957, A264226, A264412, A264413, A264415.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 20*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 A264226, then:
(1) G( x/(A(x)^2 - 16*x) ) = x,
(2) G( x/(A(x^2) + 4*x) ) = x,
(3) A(G(x))^2 = (1+16*x) * G(x)/x,
(4) A(G(x)^2) = (1-4*x) * G(x)/x,
where G(x)^2 = G( x^2/(1-8*x) ).

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) ).

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.

A271930 G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 3, 15, 90, 597, 4221, 31185, 237897, 1859568, 14816637, 119892942, 982565883, 8138777166, 68028775587, 573078135996, 4860507197700, 41470162208814, 355695498901179, 3065210379987489, 26525947283576640, 230425563258798840, 2008561878414115803, 17563090615911038115, 154014411705019299450, 1354142406561753259035, 11934928413519024726252, 105426063390991627937457, 933206579920813459523994, 8276480132736299734057275, 73535083052134446419214960
Offset: 1

Views

Author

Paul D. Hanna, Apr 16 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 + 3*x^2 + 15*x^3 + 90*x^4 + 597*x^5 + 4221*x^6 + 31185*x^7 + 237897*x^8 + 1859568*x^9 + 14816637*x^10 + 119892942*x^11 + 982565883*x^12 +...
where A(x)^2 = A( x^2 + 6*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 270*x^5 + 1959*x^6 + 14724*x^7 + 113706*x^8 + 896994*x^9 + 7198257*x^10 + 58580766*x^11 + 482345937*x^12 + 4011023556*x^13 + 33637887441*x^14 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 3*x^2 + 3*x^3 - 3*x^5 + 9*x^7 - 33*x^9 + 126*x^11 - 513*x^13 + 2214*x^15 - 9876*x^17 + 45045*x^19 - 209493*x^21 +...+ A264412(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 3*x^2 where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.

Links

Crossrefs

Cf. A264412, A271935, A271957, A271931, A271934.

Programs

  • PARI
    {a(n) = my(A=x,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 6*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) - 3*x^2 ) = x, where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.
a(n) ~ c * d^n / n^(3/2), where d = 9.35010183959428615991060685319... and c = 0.0902227396498060205291555743... . - Vaclav Kotesovec, Apr 18 2016

A274479 G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 4*x^2) ).

Original entry on oeis.org

1, 1, 4, 10, 34, 106, 361, 1219, 4252, 14932, 53263, 191533, 695233, 2540617, 9344050, 34546672, 128330533, 478653973, 1791816967, 6729202603, 25344884479, 95707901503, 362269464487, 1374203633335, 5223097370170, 19888174932226, 75856437036451, 289780169876749, 1108607284380835, 4246966803249139, 16290547536335716, 62562701811659506, 240540845892246253, 925825162823212429, 3567069859670052457, 13756707569545384033
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2016

Keywords

Comments

Compare g.f. with the identities:
(1) F(x)^2 = F( x^2/(1 - 2*x + 2*x^2) ) when F(x) = x/(1-x).
(2) M(x)^2 = M( x^2/(1 - 2*x) ) when M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).
a(n) = 1 (mod 3) for n>=1 (conjecture).
Radius of convergence of g.f. A(x) is r = 1/4 where r = r^2/(1-2*r-4*r^2) with A(1/4) = 1.
What is the limit a(n)/A000108(n) ? Note that A000108(n) = binomial(2*n,n)/(n+1) is the n-th Catalan number.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 106*x^6 + 361*x^7 + 1219*x^8 + 4252*x^9 + 14932*x^10 + 53263*x^11 + 191533*x^12 +...
such that A( x^2/(1-2*x-4*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 104*x^6 + 360*x^7 + 1306*x^8 + 4688*x^9 + 17106*x^10 + 62548*x^11 + 230570*x^12 + 853512*x^13 + 3176161*x^14 + 11866142*x^15 +...
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - x^2 - 2*x^3 + 5*x^4 + 4*x^5 - 22*x^6 - 5*x^7 + 95*x^8 - 17*x^9 - 412*x^10 + 220*x^11 + 1790*x^12 - 1559*x^13 - 7771*x^14 +...
which is related to A264412 by:
x/Series_Reversion(A(x)) = 1 + 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) +...
The g.f. G(x) of A264412 begins:
G(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 - 209493*x^10 +...
where G(x)^2 = G(x^2) + 6*x.
Also, we have A(x/(1 + x + 3*x^2))^2 = A(x^2/(1 + x^2 + 9*x^4)), where the series begin:
A(x/(1 + x + 3*x^2)) = x - 3*x^5 + 3*x^9 + 81*x^13 - 840*x^17 + 3960*x^21 + 711*x^25 - 152145*x^29 + 1009254*x^33 - 1772820*x^37 + 1991277*x^41 +...
A(x^2/(1 + x^2 + 9*x^4)) = x^2 - 6*x^6 + 15*x^10 + 144*x^14 - 2157*x^18 + 13446*x^22 - 20817*x^26 - 420876*x^30 + 4282764*x^34 - 17051652*x^38 +...
which is equal to A(x/(1 + x + 3*x^2))^2.

Links

Crossrefs

Cf. A264412, A274478, A274484.

Programs

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

Formula

G.f. A(x) satisfies: A( x/(1 + x + 3*x^2) )^2 = A( x^2/(1 + x^2 + 9*x^4) ).
Let G(x) denote the g.f. of A264412, where G(x)^2 = G(x^2) + 6*x, then g.f. A(x) satisfies:
(1) A(x) = x/(1-x) * G( A(x)^2 ),
(2) G(x^2) = x/Series_Reversion(A(x)) - x,
(3) A( x/(G(x^2) + x) ) = x,
(4) A(x)^2/(G(A(x)^4) + A(x)^2) = x^2/(1 - 2*x - 4*x^2).
Showing 1-7 of 7 results.

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