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 1-5 of 5 results.

A357547 a(n) = coefficient of x^n in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).

Original entry on oeis.org

1, 2, 9, 38, 176, 832, 4039, 19938, 99861, 506042, 2590099, 13370898, 69540016, 364028992, 1916585714, 10142059868, 53911982971, 287736310102, 1541243386819, 8282387269058, 44638363790176, 241216694913632, 1306608966475854, 7092980525443588, 38581011402034156
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2022

Keywords

Comments

Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*r^2), with A(r) = 1.
Related 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) );
here, a = 2, b = 6.

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 38*x^4 + 176*x^5 + 832*x^6 + 4039*x^7 + 19938*x^8 + 99861*x^9 + 506042*x^10 + 2590099*x^11 + 13370898*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 112*x^5 + 585*x^6 + 3052*x^7 + 16018*x^8 + 84384*x^9 + 446384*x^10 + 2370240*x^11 + 12631104*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 4*x^3 + 15*x^4 + 65*x^5 + 291*x^6 + 1356*x^7 + 6474*x^8 + 31555*x^9 + 156315*x^10 + 784924*x^11 + ... + A357785(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 5*x^2 - 10*x^4 + 50*x^6 - 305*x^8 + 2025*x^10 - 14400*x^12 + 107500*x^14 - 829415*x^16 + 6559700*x^18 - 52908950*x^20 + ...

Links

Crossrefs

Cf. A357785, A264224, A274483, A274484, A357548.

Programs

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x/(1 + 2*x + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3.a) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
(3.b) A(x)^2 = -A( -x^2/(1 - 4*x - 8*x^2) ).
(4.a) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ).
(4.b) A( x/(1 + 2*x) )^2 = -A( -x^2/(1 - 12*x^2) ).
(4.c) A( x/(1 + 2*x) )^2 = A( -x/(1 - 2*x) )^2.

A357548 a(n) = coefficient of x^n in A(x) where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).

Original entry on oeis.org

1, 2, 11, 50, 261, 1362, 7344, 40112, 222338, 1245476, 7043605, 40153390, 230518723, 1331576430, 7733934030, 45138530004, 264596552838, 1557101158092, 9195520745412, 54477134410680, 323668083179382, 1928047124332764, 11512382184408072, 68889282756213840
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2022

Keywords

Comments

Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*r^2), with A(r) = 1.
Related 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) );
here, a = 2, b = 8.

Examples

			G.f.: A(x) = x + 2*x^2 + 11*x^3 + 50*x^4 + 261*x^5 + 1362*x^6 + 7344*x^7 + 40112*x^8 + 222338*x^9 + 1245476*x^10 + 7043605*x^11 + 40153390*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 144*x^5 + 843*x^6 + 4868*x^7 + 28378*x^8 + 165664*x^9 + 971013*x^10 + 5708132*x^11 + 33660362*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + ... + A357786(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 7*x^2 - 21*x^4 + 147*x^6 - 1260*x^8 + 11907*x^10 - 120540*x^12 + 1279047*x^14 - 14029428*x^16 + 157788183*x^18 + ...

Links

Crossrefs

Cf. A357786, A264224, A274483, A274484, A357547.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 8*x^2 +x*O(x^n)) ) )
); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ).
(5) A( x/(1 + 4*x) )^2 = A( x^2/(1 + 4*x - 8*x^2) ).

A274478 G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 2*x^2) ).

Original entry on oeis.org

1, 1, 3, 7, 20, 56, 166, 498, 1530, 4762, 15022, 47862, 153859, 498239, 1623779, 5321059, 17520994, 57937106, 192304222, 640446358, 2139414409, 7166431909, 24065926653, 81003492725, 273229977460, 923438683996, 3126674842896, 10604713671208, 36025426127382, 122566140787390, 417584644921806, 1424610537707166, 4866239784751346, 16642071212737394, 56978489024931038, 195289731964727862, 670023314236521396, 2301024202252503308, 7909580344156028160
Offset: 1

Views

Author

Paul D. Hanna, Jul 26 2016

Keywords

Comments

Radius of convergence of g.f. A(x) is r = (sqrt(17) - 3)/4 where r = r^2/(1-2*r-2*r^2) with A(r) = 1.
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).

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 56*x^6 + 166*x^7 + 498*x^8 + 1530*x^9 + 4762*x^10 + 15022*x^11 + 47862*x^12 +...
such that A( x^2/(1-2*x-2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 20*x^5 + 63*x^6 + 194*x^7 + 613*x^8 + 1944*x^9 + 6236*x^10 + 20136*x^11 + 65496*x^12 + 214272*x^13 + 704774*x^14 + 2328852*x^15 +...
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - x^2 - x^3 + 3*x^4 - 7*x^6 + 4*x^7 + 15*x^8 - 16*x^9 - 32*x^10 + 51*x^11 + 69*x^12 - 153*x^13 - 148*x^14 + 445*x^15 + 315*x^16 +...
which is related to A107087 by:
x/Series_Reversion(A(x)) = 1 + 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 + x + 2*x^2))^2 = A(x^2/(1 + x^2 + 4*x^4)), where the series begin:
A(x/(1 + x + 2*x^2)) = x - x^5 - x^9 + 8*x^13 - 13*x^17 - 8*x^21 - x^25 + 307*x^29 + 135*x^33 - 9641*x^37 + 36869*x^41 +...
A(x^2/(1 + x^2 + 4*x^4)) = x^2 - 2*x^6 - x^10 + 18*x^14 - 41*x^18 - 6*x^22 + 104*x^26 + 424*x^30 - 301*x^34 - 19974*x^38 + 97752*x^42 +...
which is equal to A(x/(1 + x + 2*x^2))^2.

Links

Crossrefs

Cf. A107087, A274479, A274484, A260650.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-2*x-2*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 + 2*x^2) )^2 = A( x^2/(1 + x^2 + 4*x^4) ).
Let G(x) denote the g.f. of A107087, where G(x)^2 = G(x^2) + 4*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 - 2*x^2).

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).

A274483 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1 - 4*x - 2*x^2) ).

Original entry on oeis.org

1, 2, 8, 32, 138, 612, 2784, 12896, 60635, 288614, 1388104, 6735808, 32938438, 162156828, 803026176, 3997462368, 19991321445, 100387500906, 505950179016, 2558352514272, 12974595610122, 65975538192036, 336293496474144, 1717927441213152, 8793426613714734, 45092543870052092, 231621905868337424, 1191586088094887936, 6138909938284313524, 31668826322371245256, 163571372589617459584, 845826517521629901888, 4378463647900723645800
Offset: 1

Views

Author

Paul D. Hanna, Aug 03 2016

Keywords

Comments

Radius of convergence is r = (sqrt(33) - 5)/4 where A(r) = 1.
Compare the 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 + 8*x^3 + 32*x^4 + 138*x^5 + 612*x^6 + 2784*x^7 + 12896*x^8 + 60635*x^9 + 288614*x^10 + 1388104*x^11 + 6735808*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 + 20*x^4 + 96*x^5 + 468*x^6 + 2288*x^7 + 11248*x^8 + 55552*x^9 + 275610*x^10 + 1373192*x^11 + 6869096*x^12 +...
The series reversion of g.f. A(x) begins
Series_Reversion(A(x)) = x - 2*x^2 + 8*x^4 - 10*x^5 - 24*x^6 + 64*x^7 + 64*x^8 - 327*x^9 - 172*x^10 + 1664*x^11 + 480*x^12 - 8858*x^13 - 1328*x^14 + 49344*x^15 + 3584*x^16 - 286432*x^17 - 9714*x^18 + 1723264*x^19 + 26800*x^20 - 10669788*x^21 - 73768*x^22 + 67557440*x^23 + 200448*x^24 +...
Now compare the expansion given by
x/Series_Reversion(A(x)) = 1 + 2*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 + 20184*x^14 - 122883*x^16 + 766464*x^18 - 4875378*x^20 + 31507728*x^22 - 206278686*x^24 + 1365201252*x^26 - 9118841784*x^28 + 61393574760*x^30 - 416193047280*x^32 + 2838492444204*x^34 +...
to the series G(x) such that G(x)^2 = G(x^2) + 8*x, which begins
G(x) = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 - 122883*x^8 + 766464*x^9 - 4875378*x^10 + 31507728*x^11 - 206278686*x^12 +...
and equals the square of the g.f. of A228711.

Crossrefs

Cf. A228711, A274484, A264224.

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/(1 + 2*x + 5*x^2) )^2 = A( x^2/(1 + 4*x^2 + 25*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A( x/(1 + 2*x) ) = -A( -x/(1 - 2*x) ), an odd function.
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 6*x^2) ), an even function.
Given G(x) such that G(x)^2 = G(x^2) + 8*x, then g.f. A(x) satisfies:
(5) A(x) = x/(1-2*x) * G( A(x)^2 ),
(6) A(x) = Series_Reversion( x/(G(x)^2 - 6*x) ),
(7) G(x) = sqrt( x/Series_Reversion(A(x)) + 6*x ),
(8) G(x^2) = x/Series_Reversion(A(x)) - 2*x,
(9) A( x/(G(x)^2 - 6*x) ) = x,
(10) A( x/(G(x^2) + 2*x) ) = x,
(11) A(x)^2/(G(A(x)^4) + 2*A(x)^2) = x^2/(1 - 4*x - 2*x^2).
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.
Showing 1-5 of 5 results.

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