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G.f. A(x) satisfies: A(x)^2 = A( x^2/(1-4*x) ), with A(0) = 0.
+10
12
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
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).
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.
EXAMPLE
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...
PROG
(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), ", "))
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 12*x.
+10
7
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
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) ).
EXAMPLE
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) ).
PROG
(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), ", "))
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 20*x.
+10
6
1, 10, -45, 450, -5535, 75600, -1106100, 16953750, -268652880, 4365638550, -72354858300, 1218356280000, -20784495119850, 358457180010750, -6239532583193625, 109476057598087500, -1934128026918961515, 34378012275668994150, -614328464414815220025, 11030366153872043358750, -198899407327466712808800, 3600377821710426377668500
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) ).
EXAMPLE
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) ).
PROG
(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), ", "))
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 30*x.
+10
5
1, 15, -105, 1575, -29190, 603225, -13352850, 309605625, -7422255645, 182481301800, -4575894819300, 116581172754375, -3009161401332975, 78523515330379875, -2068113764887828875, 54904020923799337500, -1467692309121298737960, 39472725372798507822900, -1067296235915278105855650, 28996357915496677935088125, -791147023483262777604486675, 21669197341488265510394307750
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) ).
EXAMPLE
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) ).
PROG
(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), ", "))
G.f. satisfies: A(x)^2 = A( x^2/(1-6*x)^2 ).
+10
4
1, 6, 39, 270, 1959, 14706, 113166, 887004, 7050837, 56672622, 459646488, 3756181248, 30893173038, 255509028612, 2123685458190, 17728918028172, 148590381782418, 1249839423702828, 10547139497197887, 89271390230559918, 757673193636234279, 6446893091203601298, 54983813851196942292, 469959567684908644440
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.
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.
EXAMPLE
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.
PROG
(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), ", "))
G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
+10
4
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
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).
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
EXAMPLE
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.
PROG
(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), ", "))
G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 4*x^2) ).
+10
3
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
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.
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).
EXAMPLE
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 +...
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) +... 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.
PROG
(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), ", "))
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