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.

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A120593 G.f. satisfies: 5*A(x) = 4 + x + A(x)^4, starting with [1,1,6].

Original entry on oeis.org

1, 1, 6, 76, 1201, 21252, 402892, 8001412, 164321982, 3461110532, 74358814838, 1623152780808, 35897318940028, 802620009567628, 18112759482614328, 412020809942451504, 9437537418826749369, 217486633306640519124
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006, Jan 24 2008

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + x + 6*x^2 + 76*x^3 + 1201*x^4 + 21252*x^5 +...
A(x)^4 = 1 + 4*x + 30*x^2 + 380*x^3 + 6005*x^4 + 106260*x^5 +...

Crossrefs

Cf. A120588 - A120592, A120594 - A120607; A244856.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[1+5*x - (1+x)^4, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+x+6*x^2+x*O(x^n));for(i=0,n,A=A-5*A+4+x+A^4);polcoeff(A,n)}

Formula

G.f. satisfies:
(1) A(x) = 1 + Series_Reversion(1+5*x - (1+x)^4).
(2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (4+x)^(3*n+1)/5^(4*n+1), by Lagrange Inversion.
(3) A(x) = F(x/A(x)) and F(x) = A(x*F(x)) where F(x) = (4 + F(x)^4)/(5-x) is the g.f. of A244856. - Paul D. Hanna, Jul 09 2014
a(n) ~ 2^(-7/3 + 3*n) * (-32 + 15*10^(1/3))^(1/2 - n) / (5^(1/3) * n^(3/2) * sqrt(3*Pi)). - Vaclav Kotesovec, Nov 28 2017

A120594 G.f. satisfies: 8*A(x) = 7 + 8*x + A(x)^4, starting with [1,2,6].

Original entry on oeis.org

1, 2, 6, 44, 394, 3948, 42364, 476120, 5532714, 65935804, 801461012, 9897836520, 123840983812, 1566487308344, 19999112293944, 257365488659376, 3334967582746218, 43477505482249692, 569854228738577572
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 2*x + 6*x^2 + 44*x^3 + 394*x^4 + 3948*x^5 + 42364*x^6 +...
A(x)^4 = 1 + 8*x + 48*x^2 + 352*x^3 + 3152*x^4 + 31584*x^5 + 338912*x^6+..

Crossrefs

Cf. A120588 - A120593, A120595 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+8*x - (1+x)^4)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+2*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-8*A+7+8*x+A^4)/4);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+8*x - (1+x)^4)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (7+8*x)^(3*n+1)/8^(4*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(-11/6 + 3*n) * (-7 + 6*2^(1/3))^(1/2 - n) / (n^(3/2) * sqrt(3*Pi)). - Vaclav Kotesovec, Nov 28 2017

A120595 G.f. satisfies: 13*A(x) = 12 + 27*x + A(x)^4, starting with [1,3,6].

Original entry on oeis.org

1, 3, 6, 36, 249, 1932, 16044, 139500, 1253934, 11558316, 108658902, 1037800920, 10041891132, 98230257636, 969814634424, 9651213968784, 96710160474513, 974967422602428, 9881687141571732, 100632995795535588
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 249*x^4 + 1932*x^5 +...
A(x)^4 = 1 + 12*x + 78*x^2 + 468*x^3 + 3237*x^4 + 25116*x^5 +...

Crossrefs

Cf. A120588 - A120594, A120596 - A120607; A245043.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^4)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 27 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+6*x^2+x*O(x^n));for(i=0,n,A=A+(-13*A+12+27*x+A^4)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^4)/27).
G.f.: A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (12 + 27*x)^(3*n+1) / 13^(4*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(3*n - 7/3) * 3^(2*n) / (13^(1/3) * sqrt(Pi) * n^(3/2) * (2^(1/3)*13^(4/3) - 32)^(n - 1/2)). - Vaclav Kotesovec, Nov 27 2017

A120591 Self-convolution cube of A120590, such that a(n) = 4*A120590(n) for n>=2.

Original entry on oeis.org

1, 3, 12, 76, 600, 5304, 50232, 498360, 5112756, 53796820, 577370508, 6295961100, 69557631936, 776913430272, 8758443555360, 99527014659360, 1138832618425272, 13110313153525272, 151738042878341400, 1764609260161776600
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Examples

			A(x) = 1 + 3*x + 12*x^2 + 76*x^3 + 600*x^4 + 5304*x^5 + 50232*x^6 +...
A(x)^(1/3) = 1 + x + 3*x^2 + 19*x^3 + 150*x^4 + 1326*x^5 + 12558*x^6 +...

Crossrefs

Cf. A120590 (A(x)^(1/3)); A120588, A120592 - A120607.

Programs

  • PARI
    {a(n)=local(A=1+x+3*x^2+x*O(x^n));for(i=0,n,A=A-4*A+3+x+A^3);polcoeff(A^3,n)}

A120596 G.f. satisfies: 6*A(x) = 5 + x + A(x)^5, starting with [1,1,10].

Original entry on oeis.org

1, 1, 10, 210, 5505, 161601, 5082420, 167451780, 5705082795, 199354509755, 7105393162010, 257312347583330, 9440808323869455, 350189693739455535, 13110655796699158800, 494772468434359266960, 18801468275832345890970
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + x + 10*x^2 + 210*x^3 + 5505*x^4 + 161601*x^5 +...
A(x)^5 = 1 + 5*x + 60*x^2 + 1260*x^3 + 33030*x^4 + 969606*x^5 +...

Crossrefs

Cf. A120588 - A120595, A120597 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[1+6*x - (1+x)^5, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+x+10*x^2+x*O(x^n));for(i=0,n,A=A-6*A+5+x+A^5);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion(1+6*x - (1+x)^5). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (5+x)^(4*n+1)/6^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-5 + 4*(6/5)^(5/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120597 G.f. satisfies: 9*A(x) = 8 + 8*x + A(x)^5, starting with [1,2,10].

Original entry on oeis.org

1, 2, 10, 120, 1770, 29208, 516180, 9554640, 182867970, 3589443160, 71861735660, 1461730482160, 30123451315620, 627598216410480, 13197173403868200, 279728425129963680, 5970277970921643570, 128199003794219752920
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 1770*x^4 + 29208*x^5 +...
A(x)^5 = 1 + 10*x + 90*x^2 + 1080*x^3 + 15930*x^4 + 262872*x^5 +...

Crossrefs

Cf. A120588 - A120596, A120598 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+9*x - (1+x)^5)/8, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+2*x+10*x^2+x*O(x^n));for(i=0,n,A=A+(-9*A+8+8*x+A^5)/4);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+9*x - (1+x)^5)/8). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (8+8*x)^(4*n+1)/9^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-1 + 9*sqrt(3)/(10*5^(1/4)))^(1/2 - n) / (3^(3/4) * 5^(1/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120598 G.f. satisfies: 30*A(x) = 29 + 125*x + A(x)^5, starting with [1,5,10].

Original entry on oeis.org

1, 5, 10, 90, 825, 8445, 92820, 1066740, 12670635, 154308775, 1916370170, 24177471370, 309007779015, 3992428316835, 52059968802000, 684240882022800, 9055282215370050, 120563388411386850, 1613785688724362400
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 5*x + 10*x^2 + 90*x^3 + 825*x^4 + 8445*x^5 +...
A(x)^5 = 1 + 25*x + 300*x^2 + 2700*x^3 + 24750*x^4 + 253350*x^5 +...

Crossrefs

Cf. A120588 - A120597, A120599 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+30*x - (1+x)^5)/125, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+5*x+10*x^2+x*O(x^n));for(i=0,n,A=A+(-30*A+29+125*x+A^5)/25);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+30*x - (1+x)^5)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (29+125*x)^(4*n+1)/30^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 5^(-1/2 + 3*n) * (-29 + 24*6^(1/4))^(1/2 - n) / (2^(15/8) * 3^(3/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120599 G.f. satisfies: 13*A(x) = 12 + 32*x + A(x)^5, starting with [1,4,20].

Original entry on oeis.org

1, 4, 20, 280, 4660, 86728, 1727880, 36047280, 777470580, 17195957480, 387906427480, 8890184148560, 206419640698440, 4845319424269520, 114791477960006800, 2741248077305459040, 65915164046356799220
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 4*x + 20*x^2 + 280*x^3 + 4660*x^4 + 86728*x^5 +...
A(x)^5 = 1 + 20*x + 260*x^2 + 3640*x^3 + 60580*x^4 + 1127464*x^5 +...

Crossrefs

Cf. A120588 - A120598, A120600 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+13*x - (1+x)^5)/32, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+4*x+20*x^2+x*O(x^n));for(i=0,n,A=A+(-13*A+12+32*x+A^5)/8);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+13*x - (1+x)^5)/32). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(5*n,n)/(4*n+1) * (12+32*x)^(4*n+1)/13^(5*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 2^(-3/2 + 5*n) * (-12 + 4*(13/5)^(5/4))^(1/2 - n) / (5^(1/8) * 13^(3/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120600 G.f. satisfies: 7*A(x) = 6 + x + A(x)^6, starting with [1,1,15].

Original entry on oeis.org

1, 1, 15, 470, 18390, 805806, 37828981, 1860433080, 94614523740, 4935081398830, 262560448214031, 14193030016877406, 777315341935068820, 43039297954660894560, 2405249540028525971070, 135492504636185052358656
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + x + 15*x^2 + 470*x^3 + 18390*x^4 + 805806*x^5 +...
A(x)^6 = 1 + 6*x + 105*x^2 + 3290*x^3 + 128730*x^4 + 5640642*x^5 +...

Crossrefs

Cf. A120588 - A120599, A120601 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[1+7*x - (1+x)^6, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+x+15*x^2+x*O(x^n));for(i=0,n,A=A-7*A+6+x+A^6);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion(1+7*x - (1+x)^6). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (6+x)^(5*n+1)/7^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ (-6 + 5*(7/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 7^(2/5) * n^(3/2) * sqrt(5*Pi)). - Vaclav Kotesovec, Nov 28 2017

A120601 G.f. satisfies: 15*A(x) = 14 + 27*x + A(x)^6, starting with [1,3,15].

Original entry on oeis.org

1, 3, 15, 210, 3510, 65562, 1310901, 27446760, 594104940, 13187589690, 298555767279, 6867021319722, 160017552201780, 3769622456958720, 89628027015591870, 2148034269252052608, 51836638064282565579
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2006

Keywords

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + 3*x + 15*x^2 + 210*x^3 + 3510*x^4 + 65562*x^5 +...
A(x)^6 = 1 + 18*x + 225*x^2 + 3150*x^3 + 52650*x^4 + 983430*x^5 +...

Crossrefs

Cf. A120588 - A120600, A120602 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+15*x - (1+x)^6)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+15*x^2+x*O(x^n));for(i=0,n,A=A+(-15*A+14+27*x+A^6)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+15*x - (1+x)^6)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (14+27*x)^(5*n+1)/15^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1/2 + 3*n) * (-14 + 5*(5/2)^(6/5))^(1/2 - n) / (2^(3/5) * 5^(9/10) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
Previous Showing 11-20 of 28 results. Next

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