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.

Previous Showing 11-19 of 19 results.

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

A120602 G.f. satisfies: 31*A(x) = 30 + 125*x + A(x)^6, starting with [1,5,15].

Original entry on oeis.org

1, 5, 15, 190, 2550, 38070, 609205, 10199640, 176483340, 3130904150, 56641633455, 1040985874470, 19381240377460, 364777461207360, 6929053224018750, 132665646902812800, 2557591625106894075, 49604907701733017850
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 + 15*x^2 + 190*x^3 + 2550*x^4 + 38070*x^5 +...
A(x)^6 = 1 + 30*x + 465*x^2 + 5890*x^3 + 79050*x^4 + 1180170*x^5 +...

Crossrefs

Cf. A120588 - A120601, A120603 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+31*x - (1+x)^6)/125, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+5*x+15*x^2+x*O(x^n));for(i=0,n,A=A+(-31*A+30+125*x+A^6)/25);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+31*x - (1+x)^6)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (30+125*x)^(5*n+1)/31^(6*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 5^(-1/2 + 3*n) * (-30 + 5*(31/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 31^(2/5) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120603 G.f. satisfies: 16*A(x) = 15 + 27*x + A(x)^7, starting with [1,3,21].

Original entry on oeis.org

1, 3, 21, 399, 9135, 233709, 6400947, 183585897, 5443737390, 165536020650, 5133935821014, 161768728483362, 5164132704296202, 166660621950110526, 5428573285691233650, 178234125351736454070, 5892439158797172244515
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 + 21*x^2 + 399*x^3 + 9135*x^4 + 233709*x^5 +...
A(x)^7 = 1 + 21*x + 336*x^2 + 6384*x^3 + 146160*x^4 + 3739344*x^5 +...

Crossrefs

Cf. A120588 - A120602, A120604 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+16*x - (1+x)^7)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+21*x^2+x*O(x^n));for(i=0,n,A=A+(-16*A+15+27*x+A^7)/9);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+16*x - (1+x)^7)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(7*n,n)/(6*n+1) * (15+27*x)^(6*n+1)/16^(7*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 7^(-13/12 + 2*n) * 9^n * (-245 + 32*2^(2/3)*7^(5/6))^(1/2 - n) / (2^(8/3) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120604 G.f. satisfies: 24*A(x) = 23 + 64*x + A(x)^8, starting with [1,4,28].

Original entry on oeis.org

1, 4, 28, 616, 15820, 453208, 13894552, 445970128, 14796844588, 503423385080, 17467725995720, 615756709476272, 21990183407958584, 793912445913712496, 28928560840589374640, 1062498482335560005024, 39293868860176487815916
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 + 28*x^2 + 616*x^3 + 15820*x^4 + 453208*x^5 +...
A(x)^8 = 1 + 32*x + 672*x^2 + 14784*x^3 + 379680*x^4 + 10876992*x^5 +...

Crossrefs

Cf. A120588 - A120603, A120605 - A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+24*x - (1+x)^8)/64, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+4*x+28*x^2+x*O(x^n));for(i=0,n,A=A+(-24*A+23+64*x+A^8)/16);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+24*x - (1+x)^8)/64). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(8*n,n)/(7*n+1) * (23+64*x)^(7*n+1)/24^(8*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 4^(-1 + 3*n) * (-23 + 21*3^(1/7))^(1/2 - n) / (3^(3/7) * n^(3/2) * sqrt(7*Pi)). - Vaclav Kotesovec, Nov 28 2017

A120605 G.f. satisfies: 25*A(x) = 24 + 64*x + A(x)^9, starting with [1,4,36].

Original entry on oeis.org

1, 4, 36, 984, 31716, 1140552, 43895208, 1768717872, 73674176868, 3146885203432, 137085166193976, 6066992348458704, 272023207778276136, 12330039492509279184, 564072488005316830416, 26010805156782400648800
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 + 36*x^2 + 984*x^3 + 31716*x^4 + 1140552*x^5 +...
A(x)^9 = 1 + 36*x + 900*x^2 + 24600*x^3 + 792900*x^4 + 28513800*x^5 +...

Crossrefs

Cf. A120588 - A120604, A120606, A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+25*x - (1+x)^9)/64, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+4*x+36*x^2+x*O(x^n));for(i=0,n,A=A+(-25*A+24+64*x+A^9)/16);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+25*x - (1+x)^9)/64). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (24+64*x)^(8*n+1)/25^(9*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 4^(-1 + 3*n) * (-24 + 8*(5/3)^(9/4))^(1/2 - n) / (3^(1/8) * 5^(7/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

A120606 G.f. satisfies: 36*A(x) = 35 + 81*x + A(x)^9, starting with [1,3,12].

Original entry on oeis.org

1, 3, 12, 180, 3018, 56238, 1121484, 23406804, 504914175, 11167352013, 251879507880, 5771456609880, 133970974830420, 3143760834627420, 74454455230816008, 1777349666975945784, 42721359085344132657
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 + 12*x^2 + 180*x^3 + 3018*x^4 + 56238*x^5 +...
A(x)^9 = 1 + 27*x + 432*x^2 + 6480*x^3 + 108648*x^4 + 2024568*x^5 +...

Crossrefs

Cf. A120588 - A120605, A120607.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[(1+36*x - (1+x)^9)/81, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
  • PARI
    {a(n)=local(A=1+3*x+12*x^2+x*O(x^n));for(i=0,n,A=A+(-36*A+35+81*x+A^9)/27);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion((1+36*x - (1+x)^9)/81). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(9*n,n)/(8*n+1) * (35+81*x)^(8*n+1)/36^(9*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 3^(-1 + 4*n) * (-35 + 2^(21/4))^(1/2 - n) / (2^(23/8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017
Previous Showing 11-19 of 19 results.

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