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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A128319 G.f.: A(x) = 1+x*(1+2x*(1+3x*(...(1+n*x*(...)^3)^3...)^3)^3)^3.

Original entry on oeis.org

1, 1, 6, 66, 1034, 20790, 507600, 14546196, 478095264, 17722127700, 731393039376, 33262113690576, 1652889277811448, 89115877932595848, 5181554275409183904, 323216011162774715904, 21531610593372148573824
Offset: 0

Views

Author

Paul D. Hanna, Mar 07 2007

Keywords

Comments

(a(n)/n!)^(1/n) tends to 4.26315... - Vaclav Kotesovec, Oct 11 2020

Examples

			G.f.: A(x) = 1 + x*B(x)^3; B(x) = 1 + 2*x*C(x)^3; C(x) = 1 + 3*x*D(x)^3; D(x) = 1 + 4*x*E(x)^3; E(x) = 1 + 5*x*F(x)^3; F(x) = 1 + 6*x*G(x)^3; ...
where the respective sequences begin:
A=[1,1,6,66,1034,20790,507600,14546196,478095264,...];
B=[1,2,18,270,5454,135936,3992544,134386344,5088220200,...];
C=[1,3,36,684,16932,504540,17367840,673851600,28994802120,...];
D=[1,4,60,1380,40460,1404000,55499040,2443032360,118003755960,...];
E=[1,5,90,2430,82350,3262770,145741680,7183818180,385393611960,...];
F=[1,6,126,3906,150234,6693120,333506880,18208871856,1075094932464,...];
G=[1,7,168,5880,253064,12523896,688855104,41282607744,2661610538160,..];
H=[1,8,216,8424,401112,21833280,1314118080,85783244400,5998813428240,..];

Links

Crossrefs

Cf. A128318.

Programs

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

A268652 G.f. satisfies: A(x,y) = 1 + x*y*A(x,y+1)^2.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 9, 14, 5, 0, 64, 124, 74, 14, 0, 624, 1388, 1074, 352, 42, 0, 7736, 18964, 17292, 7520, 1588, 132, 0, 116416, 307088, 314356, 163728, 46561, 6946, 429, 0, 2060808, 5760704, 6434394, 3807910, 1311172, 266116, 29786, 1430, 0, 41952600, 122980872, 147159406, 95921164, 37846790, 9373620, 1438006, 126008, 4862, 0, 965497440, 2945806672, 3729264888, 2623904244, 1147995184, 327833296, 61731036, 7455440, 527900, 16796, 0
Offset: 0

Views

Author

Paul D. Hanna, Mar 16 2016

Keywords

Comments

Column 1 equals A128577.
Row sums equal A128318.
Main diagonal equals the Catalan numbers, A000108.

Examples

			This triangle of coefficients in g.f. A(x,y) begins:
1;
0, 1;
0, 2, 2;
0, 9, 14, 5;
0, 64, 124, 74, 14;
0, 624, 1388, 1074, 352, 42;
0, 7736, 18964, 17292, 7520, 1588, 132;
0, 116416, 307088, 314356, 163728, 46561, 6946, 429;
0, 2060808, 5760704, 6434394, 3807910, 1311172, 266116, 29786, 1430;
0, 41952600, 122980872, 147159406, 95921164, 37846790, 9373620, 1438006, 126008, 4862;
0, 965497440, 2945806672, 3729264888, 2623904244, 1147995184, 327833296, 61731036, 7455440, 527900, 16796;
0, 24786054816, 78270032288, 103887986400, 77816220888, 36954748286, 11761455804, 2565654006, 382043344, 37445610, 2195580, 58786; ...
where the g.f. A(x,y) = 1 + x*y*A(x,y+1)^2 begins:
A(x,y) = 1 + x*(y) + x^2*(2*y + 2*y^2) +
x^3*(9*y + 14*y^2 + 5*y^3) +
x^4*(64*y + 124*y^2 + 74*y^3 + 14*y^4) +
x^5*(624*y + 1388*y^2 + 1074*y^3 + 352*y^4 + 42*y^5) +
x^6*(7736*y + 18964*y^2 + 17292*y^3 + 7520*y^4 + 1588*y^5 + 132*y^6) +
x^7*(116416*y + 307088*y^2 + 314356*y^3 + 163728*y^4 + 46561*y^5 + 6946*y^6 + 429*y^7) +
x^8*(2060808*y + 5760704*y^2 + 6434394*y^3 + 3807910*y^4 + 1311172*y^5 + 266116*y^6 + 29786*y^7 + 1430*y^8) +...
RELATED TRIANGLES.
The triangle T1 of coefficients in A(x,y+1) begins:
1;
1, 1;
4, 6, 2;
28, 52, 29, 5;
276, 590, 430, 130, 14;
3480, 8240, 7142, 2902, 562, 42;
53232, 136352, 133820, 65892, 17440, 2380, 132;
955524, 2606056, 2811333, 1588813, 515738, 97246, 9949, 429;
19672320, 56489536, 65680352, 41222664, 15498120, 3613454, 514658, 41226, 1430;
456803328, 1369670752, 1692959656, 1154579428, 485522796, 131955696, 23376294, 2621102, 169766, 4862;
11810032896, 36744177952, 47799342376, 34885949644, 16033889224, 4899599348, 1016573628, 142394476, 12962360, 695860, 16796; ...
in which row sums form A128571:
[1, 2, 12, 114, 1440, 22368, 409248, 8585088, ...]
where
A(x,y+1) = 1 + x*(1 + y) + x^2*(4 + 6*y + 2*y^2) +
x^3*(28 + 52*y + 29*y^2 + 5*y^3) +
x^4*(276 + 590*y + 430*y^2 + 130*y^3 + 14*y^4) +
x^5*(3480 + 8240*y + 7142*y^2 + 2902*y^3 + 562*y^4 + 42*y^5) +
x^6*(53232 + 136352*y + 133820*y^2 + 65892*y^3 + 17440*y^4 + 2380*y^5 + 132*y^6) +
x^7*(955524 + 2606056*y + 2811333*y^2 + 1588813*y^3 + 515738*y^4 + 97246*y^5 + 9949*y^6 + 429*y^7) +...
The triangle T2 of coefficients in A(x,y)^2 begins:
1;
0, 2;
0, 4, 5;
0, 18, 32, 14;
0, 128, 270, 184, 42;
0, 1248, 2940, 2488, 928, 132;
0, 15472, 39513, 38364, 18266, 4372, 429;
0, 232832, 633296, 678712, 377332, 117430, 19776, 1430;
0, 4121616, 11800512, 13648092, 8478840, 3119480, 692086, 87112, 4862;
0, 83905200, 250768144, 308424612, 208690548, 86565216, 22913292, 3836896, 376736, 16796;
0, 1930994880, 5987236848, 7750642944, 5617656996, 2555316840, 767744018, 154465024, 20330760, 1607720, 58786; ...
in which row sums form A128577:
[1, 2, 9, 64, 624, 7736, 116416, 2060808, 41952600, ...]
where
A(x,y)^2 = 1 + x*(2*y) + x^2*(4*y + 5*y^2) +
x^3*(18*y + 32*y^2 + 14*y^3) +
x^4*(128*y + 270*y^2 + 184*y^3 + 42*y^4) +
x^5*(1248*y + 2940*y^2 + 2488*y^3 + 928*y^4 + 132*y^5) +
x^6*(15472*y + 39513*y^2 + 38364*y^3 + 18266*y^4 + 4372*y^5 + 429*y^6) +
x^7*(232832*y + 633296*y^2 + 678712*y^3 + 377332*y^4 + 117430*y^5 + 19776*y^6 + 1430*y^7) +...

Crossrefs

Cf. A128577 (column 1), A128318 (row sums), A128570, A000108 (diagonal), A128571.

Programs

  • PARI
    /* Print this triangle of coefficients in A(x,y): */
{T(n,k) = my(A=1); for(i=1,n, A = 1 + x*y*subst(A,y,y+1)^2 +x*O(x^n)); polcoeff(polcoeff(A,n,x),k,y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))
  • PARI
    /* Print triangle of coefficients in A(x,y+1): */
{T1(n,k) = my(A=1); for(i=1,n, A = 1 + x*y*subst(A,y,y+1)^2 +x*O(x^n)); polcoeff(polcoeff(subst(A,y,y+1),n,x),k,y)}
for(n=0,12, for(k=0,n, print1(T1(n,k),", "));print(""))
  • PARI
    /* Print triangle of coefficients in A(x,y)^2: */
{T2(n,k) = my(A=1); for(i=1,n, A = 1 + x*y*subst(A,y,y+1)^2 +x*O(x^n)); polcoeff(polcoeff(A^2,n,x),k,y)}
for(n=0,12, for(k=0,n, print1(T2(n,k),", "));print(""))

Formula

The g.f. of the row sums, A(x,1), equals the limit of nested squares given by
A(x,1) = 1 + x*(1 + 2*x*(1 + 3*x*(1 + 4*x*(...(1 + n*x*(...)^2)^2...)^2)^2)^2)^2.

A302657 a(n) = [x^n] 1 + x*(1 + 2*x*(1 + 3*x*(1 + 4*x*(1 + 5*x*(1 + ...)^n)^n)^n)^n)^n.

Original entry on oeis.org

1, 1, 4, 66, 2576, 181580, 20040132, 3176873014, 683004260416, 191131280146584, 67496202291859460, 29358012892996082966, 15422766301341408798384, 9628365732822661693594804, 7046590639669984518105404260, 5975695685335003337179698967230, 5813189543201787075970895280603392
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 11 2018

Keywords

Comments

(a(n) / (n-1)!^2)^(1/n) tends to 4.3002... - Vaclav Kotesovec, Apr 11 2018

Links

Crossrefs

Cf. A000142, A128318, A128319, A302686.

Programs

  • Mathematica
    Table[SeriesCoefficient[1 + x Fold[((#2 + 1) x #1 + 1)^n &, 0, Reverse[Range[n]]], {x, 0, n}], {n, 0, 16}]

A302688 Expansion of 1 + x*(1 + 2*x*(1 + 3*x*(1 + 4*x*(1 + 5*x*(1 + ...)^5)^4)^3)^2).

Original entry on oeis.org

1, 1, 2, 12, 162, 3888, 144768, 7693920, 551981520, 51355426992, 6010929609408, 864202875949440, 149698423474606080, 30747550680449611200, 7388611598645058636000, 2053517715502048081023360, 653614372412684344833419520, 236202930442590804658824312960
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 11 2018

Keywords

Comments

(a(n) / n!^2)^(1/n) tends to 1.36594... - Vaclav Kotesovec, Apr 12 2018

Links

Crossrefs

Cf. A095793, A128318, A128319, A138211, A138212.

Programs

  • Mathematica
    nmax = 17; CoefficientList[Series[1 + x Fold[((#2 + 1) x #1 + 1)^#2 &, 0, Reverse[Range[nmax]]], {x, 0, nmax}], x]

Formula

G.f. A(x) = 1 + x + 2*x^2 + 12*x^3 + 162*x^4 + 3888*x^5 + 144768*x^6 + 7693920*x^7 + 551981520*x^8 + ...

A302751 Expansion of 1 + x*(1 + 2*x^2*(1 + 3*x^3*(1 + 4*x^4*(1 + ...)^4)^3)^2).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 12, 0, 0, 18, 144, 0, 0, 432, 576, 2880, 0, 4320, 9408, 23040, 21600, 109440, 172800, 110880, 662400, 832320, 2678400, 4060800, 10296000, 9412992, 32922000, 63676800, 135734400, 263556528, 281030400, 973036800, 1906704000, 4069224000, 5184984960
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 2*x^3 + 12*x^6 + 18*x^9 + 144*x^10 + 432*x^13 + 576*x^14 + 2880*x^15 + ...

Crossrefs

Cf. A095793, A108643, A128318, A228866, A302688.

Programs

  • Mathematica
    nmax = 38; CoefficientList[Series[1 + x Fold[((#2 + 1) x^(#2 + 1) #1 + 1)^#2 &, 0, Reverse[Range[nmax]]], {x, 0, nmax}], x]
Previous Showing 11-15 of 15 results.

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