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 1-10 of 13 results. Next

A138212 G.f.: A(x) = 1 + x*(1 + x*(1 + x*(...(1 + x*(...)^(2n))...)^6)^4)^2.

Original entry on oeis.org

1, 1, 2, 9, 68, 732, 10250, 176654, 3613044, 85476720, 2295275372, 68949496421, 2290588299708, 83374406924240, 3299390271801838, 141034101443780374, 6475752407825487220, 317866884692663325892, 16609896989101220207880
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2008

Keywords

Examples

			G.f.: A(x)=1+x*B(x)^2, B(x)=1+x*C(x)^4, C(x)=1+x*D(x)^6, D(x)=1+x*E(x)^8, ...
where A(x),B(x),C(x),... are the g.f. of the sequences given below.
A=[1,1,2,9,68,732,10250,176654,3613044,85476720,...];
B=[1,1,4,30,328,4677,81888,1696086,40520620,1096342026,...];
C=[1,1,6,63,908,16311,347466,8519957,235763712,7259384208,...];
D=[1,1,8,108,1936,42110,1062416,30283824,958845640,...];
E=[1,1,10,165,3540,90550,2646522,86251140,3086189660,...];
F=[1,1,12,234,5848,172107,5725392,210342902,8410505748,...]; ...

Links

Crossrefs

Cf. A095793, A138211, A138213, A138214, A138215, A138216.

Programs

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

A138211 G.f.: A(x) = 1 + x*(1 + x*(1 + x*(...(1 + x*(...)^(2n-1))...)^5)^3)^1.

Original entry on oeis.org

1, 1, 1, 3, 18, 166, 2070, 32505, 614918, 13600671, 344202033, 9806468970, 310553772735, 10820519947581, 411338412455910, 16940944600551504, 751397442828052440, 35707884976794347170, 1810006747594245718317
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2008

Keywords

Examples

			G.f.: A(x)=1+x*B(x)^1, B(x)=1+x*C(x)^3, C(x)=1+x*D(x)^5, D(x)=1+x*E(x)^7, ...
where A(x),B(x),C(x),... are the g.f. of the sequences given below.
A=[1,1,1,3,18,166,2070,32505,614918,13600671,...];
B=[1,1,3,18,166,2070,32505,614918,13600671,344202033,...];
C=[1,1,5,45,570,9175,177836,4016810,103426120,2987875840,...];
D=[1,1,7,84,1358,26957,626871,16609768,492427321,16126773012,...];
E=[1,1,9,135,2658,62892,1712034,52281819,1762364970,64849739238,...];
F=[1,1,11,198,4598,126456,3950837,136929254,5186142291,212476739640,...];

Crossrefs

Cf. A095793, A138212, A138213, A138214, A138215, A138216.

Programs

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

A138213 G.f.: A(x) = 1 + x*(1 + x*(1 + x*(...(1 + x*(...)^(3n))...)^9)^6)^3.

Original entry on oeis.org

1, 1, 3, 21, 244, 4002, 84909, 2209947, 68121822, 2425846806, 97969327890, 4423628854404, 220806455598561, 12072207455321168, 717431790926502954, 46045783798588216767, 3174068594948910976851, 233875508656473241657578
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2008

Keywords

Examples

			G.f.: A(x)=1+x*B(x)^3, B(x)=1+x*C(x)^6, C(x)=1+x*D(x)^9, D(x)=1+x*E(x)^12,...
where A(x),B(x),C(x),... are the g.f. of the sequences given below.
A=[1,1,3,21,244,4002,84909,2209947,68121822,2425846806,...];
B=[1,1,6,69,1154,25062,665862,20869399,752900220,30714860088,...];
C=[1,1,9,144,3162,86346,2789703,103536696,4329341244,...];
D=[1,1,12,246,6700,221145,8453892,364604520,17444393868,...];
E=[1,1,15,375,12200,472875,20921433,1031067730,55735025670,...];
F=[1,1,18,531,20094,895077,45035802,2500543500,150992211456,...]; ...

Crossrefs

Cf. A095793, A138211, A138212, A138214, A138215, A138216.

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)}

A138214 G.f.: A(x) = 1 + x*(1 + x*(1 + x*(...(1 + x*(...)^(4n))...)^12)^8)^4.

Original entry on oeis.org

1, 1, 4, 38, 596, 13137, 373544, 13008184, 535947320, 25492727304, 1374588760980, 82844371459764, 5518323917106220, 402556752045926108, 31916585459440839392, 2732642735337686840152, 251267557458318511262096
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2008

Keywords

Examples

			G.f.: A(x)=1+x*B(x)^4, B(x)=1+x*C(x)^8, C(x)=1+x*D(x)^12, D(x)=1+x*E(x)^16,...
where A(x),B(x),C(x),... are the g.f. of the sequences given below.
A=[1,1,4,38,596,13137,373544,13008184,535947320,25492727304,...];
B=[1,1,8,124,2792,81462,2902528,121830916,5880235184,...];
C=[1,1,12,258,7612,278991,12084552,600710380,33615167976,...];
D=[1,1,16,440,16080,711740,36459968,2105685752,134824193120,...];
E=[1,1,20,670,29220,1517725,89938984,5933795760,429195194520,...];
F=[1,1,24,948,48056,2866962,193128768,14351122716,1159330814736,...]; ...

Crossrefs

Cf. A095793, A138211, A138212, A138213, A138215, A138216.

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)}

A138215 G.f.: A(x) = 1 + x*(1 + x*(1 + x*(...(1 + x*(...)^(5n))...)^15)^10)^5.

Original entry on oeis.org

1, 1, 5, 60, 1185, 32805, 1169626, 51021010, 2631549790, 156635460260, 10566145206715, 796523479440060, 66355853815084855, 6053343246845576335, 600137100011260447750, 64247982820612486908840
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2008

Keywords

Examples

			G.f.: A(x)=1+x*B(x)^5, B(x)=1+x*C(x)^10, C(x)=1+x*D(x)^15, D(x)=1+x*E(x)^20,...
where A(x),B(x),C(x),... are the g.f. of the sequences given below.
A=[1,1,5,60,1185,32805,1169626,51021010,2631549790,...];
B=[1,1,10,195,5520,202235,9038502,475490115,28745939090,...];
C=[1,1,15,405,15005,690165,37491378,2335884815,163755375450,...];
D=[1,1,20,690,31640,1756595,112818004,8165592610,654987108920,...];
E=[1,1,25,1050,57425,3739650,277763130,22962379750,2080527807050,...];
F=[1,1,30,1485,94360,7055580,595576506,55444469360,5610038179890,...]; ...

Crossrefs

Cf. A095793, A138211, A138212, A138213, A138214, A138216.

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)}

A138216 G.f.: A(x) = 1 + x*(1 + x*(1 + x*(...(1 + x*(...)^(6n))...)^18)^12)^6.

Original entry on oeis.org

1, 1, 6, 87, 2072, 69051, 2960496, 155190175, 9614870340, 687262107456, 55663739264928, 5037617218937667, 503778146624222544, 55164755650126969274, 6564517420892162939514, 843494176565238712267131
Offset: 0

Views

Author

Paul D. Hanna, Mar 06 2008

Keywords

Examples

			G.f.: A(x)=1+x*B(x)^6, B(x)=1+x*C(x)^12, C(x)=1+x*D(x)^18, D(x)=1+x*E(x)^24,...
where A(x),B(x),C(x),... are the g.f. of the sequences given below.
A=[1,1,6,87,2072,69051,2960496,155190175,9614870340,...];
B=[1,1,12,282,9616,424035,22794444,1441538178,104721633324,...];
C=[1,1,18,585,26088,1443708,94316940,7064386296,595172880432,...];
D=[1,1,24,996,54944,3668826,283322664,24650121400,2376215009736,...];
E=[1,1,30,1515,99640,7802145,696663576,69221991825,7536986249580,...];
F=[1,1,36,2142,163632,14708421,1492326612,166960071642,...]; ...

Crossrefs

Cf. A095793, A138211, A138212, A138213, A138214, A138215.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(j=0, n-1, A=1+x*A^(6*(n-j))); polcoeff(A, n)}

A120959 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^(2^n) )...)^8)^4)^2.

Original entry on oeis.org

1, 1, 2, 9, 84, 1540, 54522, 3734454, 498851832, 131025111932, 68094916593416, 70324929555472825, 144712913119662777792, 594305955799647611394896, 4875569433937264188593935824, 79943787791004406866072303453528
Offset: 0

Views

Author

Paul D. Hanna, Jul 28 2006

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x*B(x)^2; B(x) = 1 + x*C(x)^4; C(x) = 1 + x*D(x)^8;
D(x) = 1 + x*E(x)^16; E(x) = 1 + x*F(x)^32; ...
where the respective sequences begin:
B=[1,1,4,38,724,26385,1837224,247455640,65256486712,...];
C=[1,1,8,156,6008,436870,60346328,16118073852,8445009616488,...];
D=[1,1,16,632,48944,7110684,1956587408,1040720206536,...];
E=[1,1,32,2544,395104,114749560,63023951008,66902165283280,...];
F=[1,1,64,10208,3175104,1843872240,2023417888576,...].

Crossrefs

Cf. A234296, A095793.

Programs

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

A121587 G.f.: A(x) = 1+x*(1+x*(1+x*(...(1+x*(...)^(-n) )...)^-3)^-2)^-1.

Original entry on oeis.org

1, 1, -1, 3, -14, 87, -672, 6202, -66622, 817205, -11278833, 173092010, -2925096344, 53989582136, -1080876094507, 23331975207984, -540247838958615, 13357882578863941, -351281262266717583, 9790602495326179971, -288289868480192337409, 8942994568771904297378
Offset: 0

Views

Author

Paul D. Hanna, Aug 09 2006

Keywords

Examples

			G.f.: A(x) = 1 + x/B(x); B(x) = 1 + x/C(x)^2; C(x) = 1 + x/D(x)^3;
D(x) = 1 + x/E(x)^4; E(x) = 1 + x/F(x)^5; ...
where the respective sequences begin:
B=[1,1,-2,9,-58,472,-4584,51481,-655244,9318663,...];
C=[1,1,-3,18,-148,1491,-17496,232556,-3441024,56009937,...];
D=[1,1,-4,30,-300,3605,-49656,763968,-12920820,237676330,...];
E=[1,1,-5,45,-530,7400,-117096,2048865,-39048150,802555995,...];
F=[1,1,-6,63,-854,13587,-242928,4766594,-101163336,...].

Crossrefs

Cf. A095793 (variant).

Programs

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

A234301 E.g.f.: 1 + Integral (1 + Integral (1 + Integral (1 + Integral (1 + ...)^5 dx)^4 dx)^3 dx)^2 dx.

Original entry on oeis.org

1, 1, 2, 8, 54, 546, 7644, 140388, 3253608, 92429592, 3147053520, 126146938608, 5866848879168, 312780729436704, 18921429038592288, 1287533798347045536, 97808017722679006848, 8240098982756882179968, 765420628291191991328256, 77987441816127455405628672
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2013

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3/3! + 54*x^4/4! + 546*x^5/5! + 7644*x^6/6! + 140388*x^7/7! + 3253608*x^8/8! + 92429592*x^9/9! + 3147053520*x^10/10! + 126146938608*x^11/11! + 5866848879168*x^12/12! + 312780729436704*x^13/13! + 18921429038592288*x^14/14! + 1287533798347045536*x^15/15! +...
such that
A(x) = 1 + Integral B(x)^2 dx,
B(x) = 1 + Integral C(x)^3 dx,
C(x) = 1 + Integral D(x)^4 dx,
D(x) = 1 + Integral E(x)^5 dx,
E(x) = 1 + Integral F(x)^6 dx,
F(x) = 1 + Integral G(x)^7 dx, ...
The coefficients in these series begin:
A: [1, 1, 2, 8, 54, 546, 7644, 140388, 3253608, 92429592, ...];
B: [1, 1, 3, 18, 174, 2412, 44652, 1052664, 30551760, 1064478696, ...];
C: [1, 1, 4, 32, 404, 7164, 166560, 4852440, 171572760, 7190293320, ...];
D: [1, 1, 5, 50, 780, 16890, 474390, 16535460, 693410580, 34189099680, ...];
E: [1, 1, 6, 72, 1338, 34254, 1129596, 45937884, 2234626128, 127127805168, ...];
F: [1, 1, 7, 98, 2114, 62496, 2368464, 110207328, 6109240368, 394581185712, ...];
G: [1, 1, 8, 128, 3144, 105432, 4516512, 236792304, 14746211280, 1067014500336, ...];
H: [1, 1, 9, 162, 4464, 167454, 8002890, 466950060, 32289796260, 2588975822520, ...];
I: [1, 1, 10, 200, 6110, 253530, 13374780, 859772820, 65386201560, 5756311080360, ...]; ...
DERIVATIVES.
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^3;
A''' = 2*C^6 + 6*B*C^2*D^4;
A'''' = 18*C^5*D^4 + 12*B*C*D^8 + 24*B*C^2*D^3*E^5;
A''''' = 90*C^4*D^8 + 72*C^5*D^3*E^5 + 12*C^4*D^8 + 12*B*D^12 + 96*B*C*D^7*E^5 + 24*C^5*D^3*E^5 + 48*B*C*D^7*E^5 + 72*B*C^2*D^2*E^10 + 120*B*C^2*D^3*E^4*F^6;
A'''''' = 360*C^3*D^12 + 720*C^4*D^7*E^5 + 360*C^4*D^7*E^5 + 216*C^5*D^2*E^10 + 360*C^5*D^3*E^4*F^6 + 48*C^3*D^12 + 96*C^4*D^7*E^5 + 12*C^3*D^12 + 144*B*D^11*E^5 + 96*C^4*D^7*E^5 + 96*B*D^11*E^5 + 672*B*C*D^6*E^10 + 480*B*C*D^7*E^4*F^6 + 120*C^4*D^7*E^5 + 72*C^5*D^2*E^10 + 120*C^5*D^3*E^4*F^6 + 48*C^4*D^7*E^5 + 48*B*D^11*E^5 + 336*B*C*D^6*E^10 + 240*B*C*D^7*E^4*F^6 + 72*C^5*D^2*E^10 + 144*B*C*D^6*E^10 + 144*B*C^2*D*E^15 + 720*B*C^2*D^2*E^9*F^6 + 120*C^5*D^3*E^4*F^6 + 240*B*C*D^7*E^4*F^6 + 360*B*C^2*D^2*E^9*F^6 + 480*B*C^2*D^3*E^3*F^12 + 720*B*C^2*D^3*E^4*F^5*G^7; ...
and then evaluate at x=0, where 1=A(0)=B(0)=C(0)=D(0)=E(0)=...

Links

Crossrefs

Cf. A095793, A234296.

Programs

  • PARI
    {a(n) = my(A=1); for(k=0,n-1, A = 1 + intformal((A+x*O(x^n))^(n+1-k))); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Print table of related series A,B,C,D,E,... */
{a(n, r=1) = my(A=vector(3*n+2*r+2, i, 1+x));
for(m=1, 2*n+r, for(j=0, n+r+m, A[n+r+m-j+1] = 1 + intformal((A[n+r+m-j+2] + x^r*O(x^n))^(n+r+m-j+2)) ); ); polcoeff(A[r], n)}
for(r=1,10, for(n=0, 10, print1(n!*a(n, r), ", "));print(""))
/* Print this sequence (at row r=1): */
for(n=0, 25, print1(n!*a(n, 1), ", "))

A228866 G.f.: A(x) = 1 + x*B(x), where B(x) = 1 + x^2*C(x)^2, C(x) = 1 + x^3*D(x)^3, D(x) = 1 + x^4*E(x)^4, ...

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 6, 0, 0, 6, 6, 24, 0, 15, 26, 48, 36, 140, 120, 60, 288, 279, 600, 660, 1476, 822, 2166, 2880, 5100, 7047, 6300, 14100, 21440, 30210, 30054, 62496, 72060, 123180, 174780, 253980, 319488, 497544, 730560, 976020, 1654856, 1997706, 3085932, 4160740, 6426480
Offset: 0

Views

Author

Paul D. Hanna, Sep 06 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + x^3 + 2*x^6 + x^9 + 6*x^10 + 6*x^13 + 6*x^14 + 24*x^15 +...
where A(x) = 1 + x*B(x),
B(x) = 1 + x^2 + 2*x^5 + x^8 + 6*x^9 + 6*x^12 + 6*x^13 + 24*x^14 +...
B(x) = 1 + x^2*C(x)^2,
C(x) = 1 + x^3 + 3*x^7 + 3*x^11 + 12*x^12 + x^15 + 24*x^16 + 18*x^17 +...
C(x) = 1 + x^3*D(x)^3,
D(x) = 1 + x^4 + 4*x^9 + 6*x^14 + 20*x^15 + 4*x^19 + 60*x^20 + 40*x^21 +...
D(x) = 1 + x^4*E(x)^4,
E(x) = 1 + x^5 + 5*x^11 + 10*x^17 + 30*x^18 + 10*x^23 + 120*x^24 + 75*x^25 +...
E(x) = 1 + x^5*F(x)^5,
F(x) = 1 + x^6 + 6*x^13 + 15*x^20 + 42*x^21 + 20*x^27 + 210*x^28 + 126*x^29 +...
F(x) = 1 + x^6*G(x)^6,
G(x) = 1 + x^7 + 7*x^15 + 21*x^23 + 56*x^24 + 35*x^31 + 336*x^32 + 196*x^33 +...
G(x) = 1 + x^7*H(x)^7,
H(x) = 1 + x^8 + 8*x^17 + 28*x^26 + 72*x^27 + 56*x^35 + 504*x^36 + 288*x^37 +...
...

Links

Crossrefs

Cf. A108643, A095793.

Programs

  • PARI
    {a(n)=local(A=1);for(k=1,n,A = 1 + (x*A)^(n-k+1) +x*O(x^n));polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))
Showing 1-10 of 13 results. Next

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