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.

A234296 E.g.f.: 1 + Integral (1 + Integral (1 + Integral (1 + Integral (1 + ...)^16 dx)^8 dx)^4 dx)^2 dx.

Original entry on oeis.org

1, 1, 2, 10, 112, 2544, 110944, 9088160, 1395985024, 405640228736, 225812739686144, 243825339649539840, 515865727833142919168, 2154502537039937189822464, 17852312368540223401725132800, 294428418578798287467609655705600, 9684259826489059207872454620228222976
Offset: 0

Views

Author

Paul D. Hanna, Dec 24 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 10*x^3/3! + 112*x^4/4! + 2544*x^5/5! +...
such that
A(x) = 1 + Integral B(x)^2 dx,
B(x) = 1 + Integral C(x)^4 dx,
C(x) = 1 + Integral D(x)^8 dx,
D(x) = 1 + Integral E(x)^16 dx,
E(x) = 1 + Integral F(x)^32 dx,
F(x) = 1 + Integral G(x)^64 dx, ...
The coefficients in these series begin:
A: [1, 1, 2, 10, 112, 2544, 110944, 9088160, 1395985024, ...];
B: [1, 1, 4, 44, 1048, 48472, 4171008, 663109888, 196890206720, ...];
C: [1, 1, 8, 184, 9040, 845712, 144855616, 45401856704, ...];
D: [1, 1, 16, 752, 75040, 14126752, 4830297984, 3006883867264, ...];
E: [1, 1, 32, 3040, 611392, 230931264, 157795465984, ...];
F: [1, 1, 64, 12224, 4935808, 3734695552, 5101948036608, ...];
G: [1, 1, 128, 49024, 39665920, 60075785472, 164109335366656, ...];
H: [1, 1, 256, 196352, 318046720, 963787028992, 5265107899521024, ...]; ...
To illustrate a(n) = d^n/dx^n A(x) at x=0, take successive derivatives of A=A(x):
A' = B^2;
A'' = 2*B*C^4;
A''' = 2*C^8 + 8*B*C^3*D^8;
A'''' = 24*C^7*D^8 + 24*B*C^2*D^16 + 64*B*C^3*D^7*E^16; ...
and then evaluate at x=0, where 1=A(0)=B(0)=C(0)=D(0)=E(0)=...

Links

Crossrefs

Cf. A120959, A234301.

Programs

  • PARI
    {a(n)=local(A=1); for(k=0, n-1, A=1+intformal((A+x*O(x^n))^(2^(n-k)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

A184575 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^(3^n) )...)^27)^9)^3.

Original entry on oeis.org

1, 1, 3, 30, 892, 76554, 19138212, 14126533902, 31053145918644, 204151364083796877, 4021430292908836847748, 237530957105884844479669995, 42082478775006270167542801189164, 22365250673182738144111737076795384386
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2011

Keywords

Comments

Limit a(n)/3^[n(n-1)/2] = 1.361839192264541770366149558100...

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 30*x^3 + 892*x^4 + 76554*x^5 +...
Related functions are defined by:
A(x) = 1 + x*B(x)^3;
B(x) = 1 + x*C(x)^9;
C(x) = 1 + x*D(x)^27;
D(x) = 1 + x*E(x)^81;
E(x) = 1 + x*F(x)^243; ...
where the coefficients in the above functions begin:
B=[1,1,9,279,24870,6324282,4695640434,10341522771762,...];
C=[1,1,27,2538,678708,515666952,1144737153180,7549554318496218,...];
D=[1,1,81,22923,18390510,41861447352,278471836036890,...];
E=[1,1,243,206550,497133612,3393278306694,67693048457727060,...];
F=[1,1,729,1859679,13427919990,274923122390262,16451387497191947778,...].

Crossrefs

Cf. A120959, A184576, A184577.

Programs

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

A184576 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^(4^n) )...)^64)^16)^4.

Original entry on oeis.org

1, 1, 4, 70, 4772, 1256737, 1300189840, 5343387957552, 87635577037531120, 5745017883201162928720, 1506152833843572741772875256, 1579354544764797425414872390884936, 6624338541974058042578557252396267711144
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2011

Keywords

Comments

Limit a(n)/4^[n(n-1)/2] = 1.2167020033386380312738338894882216...

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 70*x^3 + 4772*x^4 + 1256737*x^5 +...
Related functions are defined by:
A(x) = 1 + x*B(x)^4;
B(x) = 1 + x*C(x)^16;
C(x) = 1 + x*D(x)^64;
D(x) = 1 + x*E(x)^256;
E(x) = 1 + x*F(x)^1024; ...
where the coefficients in the above functions begin:
B=[1,1,16,1144,310320,324057372,1334856912400,21904872061736520,...];
C=[1,1,64,18400,19940032,83109401072,1367783819781952,...];
D=[1,1,256,294784,1277438720,21285675278272,1400838812478999808,...];
E=[1,1,1024,4718080,81776520192,5449752029298432,1434517374576707974144,...];
F=[1,1,4096,75495424,5234024427520,1395176151497833472,...].

Crossrefs

Cf. A120959, A184575, A184577.

Programs

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

A184577 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^(5^n) )...)^125)^25)^5.

Original entry on oeis.org

1, 1, 5, 135, 17635, 11196380, 35146865626, 549810415675025, 42966178319025765725, 16784792206658535573353275, 32783305434744311217446987595100, 320150590803319511079060107920058643150
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2011

Keywords

Comments

Limit a(n)/5^[n(n-1)/2] = 1.153465224918418179626705381295781...

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 135*x^3 + 17635*x^4 + 11196380*x^5 +...
Related functions are defined by:
A(x) = 1 + x*B(x)^5;
B(x) = 1 + x*C(x)^25;
C(x) = 1 + x*D(x)^125;
D(x) = 1 + x*E(x)^625;
E(x) = 1 + x*F(x)^3125; ...
where the coefficients in the above functions begin:
B=[1,1,25,3425,2224175,7020109525,109933742396880,8592795153472287100,...];
C=[1,1,125,85875,278520875,4390379300750,343605429779453150,...];
D=[1,1,625,2148125,34827604375,2744338779394375,1073806035818802734500,...];
E=[1,1,3125,53709375,4353763021875,1715255710695800000,...];
F=[1,1,15625,1342765625,544228190109375,1072040316107238765625,...].

Crossrefs

Cf. A120959, A184575, A184576.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(j=0, n-1, A=1+x*A^(5^(n-j))); polcoeff(A, n)}

A121588 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^(-2^n) )...)^-4)^-2)^-1.

Original entry on oeis.org

1, 1, -1, 3, -16, 145, -2347, 72498, -4459887, 554300965, -139235329270, 70475747813447, -71685052573258824, 146249172542467865074, -597744865134782025119044, 4890851047359454263328433041, -80078758027845307168595201926254
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2006

Keywords

Comments

Limit |a(n)|/2^[(n-1)*(n-2)/2] = 1.97254925752982255...

Examples

			G.f.: A(x) = 1 + x/B(x); B(x) = 1 + x/C(x)^2; C(x) = 1 + x/D(x)^4;
D(x) = 1 + x/E(x)^8; E(x) = 1 + x/F(x)^16; ...
where the respective sequences begin:
B=[1,1,-2,11,-112,2025,-67324,4305909,-545113744,...];
C=[1,1,-4,42,-836,30259,-2041616,265712044,-68214603840,...];
D=[1,1,-8,164,-6456,467850,-63614840,16702037652,...];
E=[1,1,-16,648,-50736,7358500,-2008876560,1059405119352,...];
F=[1,1,-32,2576,-402272,116732040,-63860549280,...].

Crossrefs

Cf. A120959 (variant).

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(j=0, n, A=1+x/A^(2^(n-j))); polcoeff(A, n)}
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.