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 1-4 of 4 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) ).

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) ).

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.
Showing 1-4 of 4 results.

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