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.

A119820 Coefficients of x^n in the n-th iteration of x*(1+x)^2 for n>=1.

Original entry on oeis.org

1, 4, 27, 300, 4790, 101010, 2660028, 84191772, 3115739358, 132074618544, 6311492388432, 335744715016854, 19678501474466211, 1260060524755139120, 87519840721085385096, 6553840567691077634748, 526360263009035464610574
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2006

Keywords

Examples

			The successive iterations of F(x) = x*(1+x)^2 begin:
F(x) = (1)x + 2x^2 + x^3
F(F(x)) = x + (4)x^2 + 10x^3 + 18x^4 + 23x^5 + 22x^6 + 15x^7 + 6x^8 +...
F(F(F(x))) = x + 6x^2 + (27)x^3 + 102x^4 + 333x^5 + 960x^6 + 2472x^7 +...
F(F(F(F(x)))) = x + 8x^2 + 52x^3 + (300)x^4 + 1578x^5 + 7692x^6 +...
F(F(F(F(F(x))))) = x + 10x^2 + 85x^3 + 660x^4 + (4790)x^5 + 32920x^6+...
F(F(F(F(F(F(x)))))) = x + 12x^2 +126x^3 +1230x^4+11385x^5+(101010)x^6+...

Crossrefs

Cf. A119821, A112317, A119819, A119817, A119815.

Programs

  • PARI
    {a(n)=local(F=x*(1+x)^2, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n, x)))}

Formula

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F_1(x)) with F_1(x) = x*(1+x)^2.

A119821 Coefficients of x^n in the n-th iteration of x/(1-x)^2 for n>=1.

Original entry on oeis.org

1, 4, 33, 436, 8015, 189596, 5494797, 188692708, 7494744807, 338103170428, 17079035749061, 955117390512858, 58584586487137113, 3910851585418994256, 282272352712037938081, 21904366942822876046020
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2006

Keywords

Comments

The coefficient of x^n in the n-th iteration of x/(1-x) = n^(n-1) = A000169(n); does this variant have a simple formula for a(n)?

Examples

			The successive iterations of F(x) = x/(1-x)^2 begin:
F(x) = (1)x + 2x^2 + 3x^3 + 4x^4 + 5x^5 + 6x^6 + 7x^7 + 8x^8 +...
F(F(x)) = x + (4)x^2 + 14x^3 + 46x^4 + 145x^5 + 444x^6 + 1331x^7 +...
F(F(F(x))) = x + 6x^2 + (33)x^3 + 174x^4 + 892x^5 + 4480x^6 +...
F(F(F(F(x)))) = x + 8x^2 + 60x^3 + (436)x^4 + 3102x^5 + 21728x^6 +...
F(F(F(F(F(x))))) = x + 10x^2 + 95x^3 + 880x^4 + (8015)x^5 +72090x^6+..
F(F(F(F(F(F(x)))))) = x + 12x^2+138x^3+1554x^4+17255x^5+(189596)x^6+..

Crossrefs

Cf. A119820, A112317; A119819, A119817, A119815.

Programs

  • PARI
    {a(n)=local(F=x/(1-x)^2, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n, x)))}

Formula

a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F(x)) with F(x) = x/(1-x)^2.

A153389 a(n) is the coefficient of x^n in the (n+1)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.

Original entry on oeis.org

1, 1, 3, 24, 340, 6950, 185332, 6102096, 240066324, 11049803253, 586427736400, 35496295553400, 2429241782433336, 186578739919860938, 15978032482994396274, 1516714259732732880752, 158739955140174552148004
Offset: 1

Views

Author

Paul D. Hanna, Jan 04 2009

Keywords

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 24*x^4 + 340*x^5 + 6950*x^6 +...
The initial coefficients of successive iterations of A(x) are as follows:
[1,1,3,24,340,6950,185332,6102096,240066324,11049803253,...];
[(1),2,8,64,866,17042,442452,14294645,554552353,25248014932,...];
[1,(3),15,126,1666,31676,799438,25293855,966179243,...];
[1,4,(24),216,2852,52800,1295784,40085834,1505339722,...];
[1,5,35,(340),4560,83030,1986288,60029146,2212932714,...];
[1,6,48,504,(6950),125770,2945660,86988543,3144212187,...];
[1,7,63,714,10206,(185332),4273850,123504913,4373929873,...];
[1,8,80,976,14536,267056,(6102096),173006484,6003091828,...];
[1,9,99,1296,20172,377430,8599692,(240066324),8167689120,...];
[1,10,120,1680,27370,524210,11981476,330711177,(11049803253),...];...
where the diagonal of terms in parenthesis form this sequence.
Some explicit expansions:
A(A(x)) = x + 2*x^2 + 8*x^3 + 64*x^4 + 866*x^5 + 17042*x^6 +...;
A(A(A(x))) = x + 3*x^2 + 15*x^3 + 126*x^4 + 1666*x^5 + 31676*x^6 +...;
A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 216*x^4 + 2852*x^5 + 52800*x^6 +...;

Crossrefs

Cf. A119819.

Programs

  • PARI
    {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}

A153849 a(n) is the coefficient of x^n in the (n+2)-th self-composition of g.f. A(x) for n>=1, with a(1)=1.

Original entry on oeis.org

1, 1, 4, 40, 675, 16023, 488264, 18150408, 798091737, 40691618165, 2372928554920, 156676126511294, 11619829396974162, 961567152709817189, 88268869225674860344, 8941327301073770038304, 994670849523023025795108
Offset: 1

Views

Author

Paul D. Hanna, Jan 04 2009

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 40*x^4 + 675*x^5 + 16023*x^6 +...
Coefficients of successive iterations of A(x) are as follows:
[1,1,4,40,675,16023,488264,18150408,798091737,40691618165,...];
[1,2,10,101,1658,38327,1144412,41900873,1821202358,...];
[(1),3,18,189,3063,69204,2024472,72937181,3130859853,...];
[1,(4),28,310,5028,111648,3202916,113479712,4806652072,...];
[1,5,(40),470,7715,169475,4777592,166446886,6951781033,...];
[1,6,54,(675),11310,247443,6875376,235657813,9700212066,...];
[1,7,70,931,(16023),351372,9658544,326079187,13225766705,...];
[1,8,88,1244,22088,(488264),13331864,444121464,17753548560,...];
[1,9,108,1620,29763,666423,(18150408),597989364,23574131721,...];
[1,10,130,2065,39330,895575,24428084,(798091737),31060981566,...];
[1,11,154,2585,51095,1186988,32546888,1057515833,(40691618165),...];
where the diagonal of terms in parenthesis form this sequence.
Some explicit expansions:
A(A(x)) = x + 2*x^2 + 10*x^3 + 101*x^4 + 1658*x^5 + 38327*x^6 +...;
A(A(A(x))) = x + 3*x^2 + 18*x^3 + 189*x^4 + 3063*x^5 + 69204*x^6 +...;
A(A(A(A(x)))) = x + 4*x^2 + 28*x^3 + 310*x^4 + 5028*x^5 + 111648*x^6 +...

Crossrefs

Cf. A119819, A153389.

Programs

  • PARI
    {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}

A163594 a(n+1) equals the coefficient of x^n in the 2^(n-1)-th iteration of g.f. A(x) = Sum_{m>=1} a(m)*x^m for n>=1 with a(1)=1.

Original entry on oeis.org

1, 1, 2, 20, 804, 108304, 49833296, 87606851264, 641794234287360, 19783636266156204928, 2512584289692759254055168, 1295158553795409705964052724736, 2690610592205668589191756477437574144
Offset: 1

Views

Author

Paul D. Hanna, Aug 10 2009

Keywords

Examples

			The coefficients of the 2^(n-1)-th iterations of the g.f. begin:
(1),1,2,20,804,108304,49833296,87606851264,641794234287360,...
1,(2),6,51,1750,222706,100558052,175666197420,1284466715882828,...
1,4,(20),170,4340,474238,204872756,353171251288,2572462315656538,...
1,8,72,(804),15560,1128036,426923128,713954691088,5159170997828364,...
1,16,272,5000,(108304),4271464,962562608,1461234395040,...
1,32,1056,35856,1266720,(49833296),3774562656,3128786120000,...
1,64,4160,273440,18169920,1226585248,(87606851264),12455033590400,...
1,128,16512,2140224,278454400,36359377216,4771446963584,(641794234287360),...
in which the main diagonal forms this sequence shift left.

Crossrefs

Cf. A119819.

Programs

  • PARI
    {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n-1, G=subst(F, x, G);F=G); return(polcoeff(G, n-1, x))))}
Showing 1-5 of 5 results.

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