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.

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.

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) ).

A271935 G.f. A(x) satisfies: A(x) = A( x^2 + 8*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 4, 26, 200, 1691, 15204, 142710, 1382568, 13721765, 138802136, 1425785270, 14832383488, 155947271878, 1654494195340, 17690004381000, 190426309700616, 2062071992480208, 22447191471665160, 245501068961175090, 2696300196714320520, 29725402250477117175, 328835072363241763920
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 + 4*x^2 + 26*x^3 + 200*x^4 + 1691*x^5 + 15204*x^6 + 142710*x^7 + 1382568*x^8 + 13721765*x^9 + 138802136*x^10 + 1425785270*x^11 + ...
where A(x)^2 = A( x^2 + 8*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 608*x^5 + 5658*x^6 + 54336*x^7 + 534984*x^8 + 5373824*x^9 + 54866075*x^10 + 567775856*x^11 + 5942353444*x^12 + ...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 4*x^2 + 6*x^3 - 15*x^5 + 90*x^7 - 660*x^9 + 5310*x^11 - 45765*x^13 + 413640*x^15 - 3864345*x^17 + 37014120*x^19 + ... + A264413(n)*x^(2*n+1) + ...
such that B(x) = x*G(x^2) - 4*x^2 where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
From _Paul D. Hanna_, May 20 2024: (Start)
The series (A(x)/x)^(1/4) seems to consist solely of integer coefficients
(A(x)/x)^(1/4) = 1 + x + 5*x^2 + 34*x^3 + 268*x^4 + 2305*x^5 + 20988*x^6 + 198891*x^7 + 1941111*x^8 + 19377707*x^9 + 196936775*x^10 + ...
and continues to be integral for at least the initial 400 coefficients. (End)

Links

Crossrefs

Cf. A264413, A271930, A271957, 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 + 8*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) - 4*x^2 ) = x, where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.

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

Views

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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