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 15 results. Next

A215703 A(n,k) is the n-th derivative of f_k at x=1, and f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways; square array A(n,k), n>=0, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 2, 12, 8, 0, 1, 1, 6, 9, 52, 10, 0, 1, 1, 4, 27, 32, 240, 54, 0, 1, 1, 2, 18, 156, 180, 1188, -42, 0, 1, 1, 2, 15, 100, 1110, 954, 6804, 944, 0, 1, 1, 8, 9, 80, 650, 8322, 6524, 38960, -5112, 0, 1, 1, 6, 48, 56, 590, 4908, 70098, 45016, 253296, 47160, 0
Offset: 0

Views

Author

Alois P. Heinz, Aug 21 2012

Keywords

Comments

A000081(m) distinct functions are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways. Some functions are representable in more than one way, the number of valid parenthesizations is A000108(m-1). The f_k are ordered, such that the number m of x's in f_k is a nondecreasing function of k. The exact ordering is defined by the algorithm below.
The list of functions f_1, f_2, ... begins:
| f_k : m : function (tree) : representation(s) : sequence |
+-----+---+------------------+--------------------------+----------+
| f_1 | 1 | x -> x | x | A019590 |
| f_2 | 2 | x -> x^x | x^x | A005727 |
| f_3 | 3 | x -> x^(x*x) | (x^x)^x | A215524 |
| f_4 | 3 | x -> x^(x^x) | x^(x^x) | A179230 |
| f_5 | 4 | x -> x^(x*x*x) | ((x^x)^x)^x | A215704 |
| f_6 | 4 | x -> x^(x^x*x) | (x^x)^(x^x), (x^(x^x))^x | A215522 |
| f_7 | 4 | x -> x^(x^(x*x)) | x^((x^x)^x) | A215705 |
| f_8 | 4 | x -> x^(x^(x^x)) | x^(x^(x^x)) | A179405 |

Examples

			Square array A(n,k) begins:
  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,   1,    1,    1,     1,     1,     1,     1, ...
  0,   2,    4,    2,     6,     4,     2,     2, ...
  0,   3,   12,    9,    27,    18,    15,     9, ...
  0,   8,   52,   32,   156,   100,    80,    56, ...
  0,  10,  240,  180,  1110,   650,   590,   360, ...
  0,  54, 1188,  954,  8322,  4908,  5034,  2934, ...
  0, -42, 6804, 6524, 70098, 41090, 47110, 26054, ...

Links

Crossrefs

Columns k=1-17, 37 give: A019590, A005727, A215524, A179230, A215704, A215522, A215705, A179405, A215706, A215707, A215708, A215709, A215691, A215710, A215643, A215629, A179505, A211205.
Rows n=0+1, 2-10 give: A000012, A215841, A215842, A215834, A215835, A215836, A215837, A215838, A215839, A215840.
Number of distinct values taken for m x's by derivatives n=1-10: A000012, A028310, A199085, A199205, A199296, A199883, A215796, A215971, A216062, A216403.
Main diagonal gives A306739.
Cf. A000081, A000108, A033917, A211192, A214569, A214570, A214571, A216041, A216281, A216349, A216350, A216351, A216368, A222379, A222380, A277537, A306679, A306710, A306726.

Programs

  • Maple
    T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
      nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
    end():
A:= (n, k)-> n!*coeff(series(subs(x=x+1, f(k)), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);
  • Mathematica
    T[n_] := If[n == 1, {x}, Map[x^#&, g[n - 1, n - 1]]];
g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten @ Table[ Table[ Table[ Product[T[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {v, g[n - i*j, i - 1]}], {w, Subsets[ Range[ Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];
f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];
A[n_, k_] := n! * SeriesCoefficient[f[k] /. x -> x+1, {x, 0, n}];
Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 08 2019, after Alois P. Heinz *)

A215522 n-th derivative of (x^x)^(x^x) at x=1.

Original entry on oeis.org

1, 1, 4, 18, 100, 650, 4908, 41090, 382520, 3863808, 42409440, 497972112, 6259762320, 83343114504, 1175904241848, 17442325040520, 272149555445760, 4438451554802880, 75714874759039104, 1343817666163911168, 24837691533530152320, 475811860099666527360
Offset: 0

Views

Author

Alois P. Heinz, Aug 14 2012

Keywords

Comments

Also n-th derivative of (x^(x^x))^x = x^(x^x*x) at x=1.
First term < 0: a(65).

Links

Crossrefs

Column k=6 of A215703.
Cf. A005727, A179230, A179405, A215524.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^(x^x) ), x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    m = 21; CoefficientList[(x+1)^((x+1)^(x+2)) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

Formula

E.g.f.: (x+1)^((x+1)^(x+2)).

A215704 n-th derivative of ((x^x)^x)^x at x=1.

Original entry on oeis.org

1, 1, 6, 27, 156, 1110, 8322, 70098, 646272, 6333336, 66712680, 745731360, 8780828328, 108873486072, 1413807287760, 19157627737080, 270460073295360, 3965693824244160, 60266513065134528, 947644484349584448, 15389579447794454400, 257702782790624613120
Offset: 0

Views

Author

Alois P. Heinz, Aug 21 2012

Keywords

Comments

Also n-th derivative of x^(x^3) at x=1.
First term < 0: a(57).

Links

Crossrefs

Column k=5 of A215703.
Cf. A005727, A179230, A179405, A215524.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, ((x^x)^x)^x ), x, n+1), x, n):
seq(a(n), n=0..25);

Formula

E.g.f.: (x+1)^((x+1)^3).

A211205 n-th derivative of x^(x^(x^(x^(x^x)))) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 60494, 823528, 12365424, 206078880, 3745686912, 74083090872, 1579529362944, 36165466533000, 884104045301640, 22992315801392064, 633547543117707648, 18439576158792912192, 565162707747635408448, 18194047307015185486080
Offset: 0

Views

Author

Alois P. Heinz and Robert G. Wilson v, Feb 04 2013

Keywords

Links

Crossrefs

Cf. A005727, A008405, A176118, A179230, A179405, A179505, A215522, A215524, A215629, A215643, A215691, A215704, A215705, A215706, A215707, A215708, A215709, A215710, A215522, A295106.
Column k=37 of A215703.
Column k=6 of A277537.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^(x^(x^x))))), x, n+1), x, n):
seq(a(n), n=0..20);
  • Mathematica
    NestList[ Factor[ D[#1, x]] &, x^x^x^x^x^x, 9] /. (x -> 1) (* or quicker *)
Range[0, 20]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 20}], x]

Formula

E.g.f.: (x+1)^((x+1)^((x+1)^((x+1)^((x+1)^(x+1))))).

A215643 n-th derivative of x^((x^(x^x))^x) at x=1.

Original entry on oeis.org

1, 1, 2, 15, 104, 890, 8814, 100660, 1288048, 18337680, 286674960, 4882660464, 89880715704, 1777384045944, 37552294300416, 843830334815640, 20086549955304384, 504750167170162944, 13348550475903813120, 370499740676381737728, 10766442934111876381440
Offset: 0

Views

Author

Alois P. Heinz, Aug 18 2012

Keywords

Comments

Also n-th derivative of x^((x^x)^(x^x)) = x^(x^(x^x*x)) at x=1.

Links

Crossrefs

Column k=15 of A215703.
Cf. A005727, A179230, A179405, A215522, A215524.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, x^((x^(x^x))^x) ), x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    m = 20;
CoefficientList[(x+1)^(((x+1)^((x+1)^(x+1)))^(x+1)) + O[x]^(m+1), x]* Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

Formula

E.g.f.: (x+1)^(((x+1)^((x+1)^(x+1)))^(x+1)).

A293472 Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^x, evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 8, 12, 12, 4, 1, 10, 40, 30, 20, 5, 1, 54, 60, 120, 60, 30, 6, 1, -42, 378, 210, 280, 105, 42, 7, 1, 944, -336, 1512, 560, 560, 168, 56, 8, 1, -5112, 8496, -1512, 4536, 1260, 1008, 252, 72, 9, 1
Offset: 0

Views

Author

Peter Luschny, Oct 10 2017

Keywords

Examples

			Triangle starts:
0: [  1]
1: [  1,   1]
2: [  2,   2,   1]
3: [  3,   6,   3,   1]
4: [  8,  12,  12,   4,   1]
5: [ 10,  40,  30,  20,   5,  1]
6: [ 54,  60, 120,  60,  30,  6, 1]
7: [-42, 378, 210, 280, 105, 42, 7, 1]
...
For n = 3, the 3rd derivative of x^x is p(3,x,t) = x^x*t^3 + 3*x^x*t^2 + 3*x^x*t + x^x + 3*x^x*t/x + 3*x^x/x - x^x/x^2 where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 3 + 6*t + 3*t^2 + t^3 with coefficients [3, 6, 3, 1].

Crossrefs

More generally, consider the n-th derivative of x^(x^m). This is case m = 1.
m | t = -1 | t = 0 | t = 1 | p(n, t) | related
m = 1 | A203852 | A005727 | A293297 | A293472 | A293239
m = 2 | - | A215524 | - | A293473 | A290268
m = 3 | - | A215704 | - | A293474 | -
Cf. A215703.

Programs

  • Maple
    dx := proc(m, n) if n = 0 then return [1] fi;
subs(ln(x) = t, diff(x^(x^m), x$n)): subs(x = 1, %):
PolynomialTools:-CoefficientList(%,t) end:
ListTools:-Flatten([seq(dx(1, n), n=0..10)]);
  • Mathematica
    dx[m_, n_] := ReplaceAll[CoefficientList[ReplaceAll[Expand[D[x^x^m, {x, n}]], Log[x] -> t], t], x -> 1];
Table[dx[1, n], {n, 0, 7}] // Flatten

A215629 n-th derivative of x^(x^((x^x)^x)) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 80, 660, 6714, 77084, 1005640, 14572944, 233086920, 4066783512, 76906345944, 1566049091568, 34153725715368, 793996577407560, 19595885746343808, 511550462381982528, 14080034085212120256, 407434430977558009344, 12363449947108075756800
Offset: 0

Views

Author

Alois P. Heinz, Aug 18 2012

Keywords

Links

Crossrefs

Column k=16 of A215703.
Cf. A005727, A179230, A179405, A215522, A215524.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, x^(x^((x^x)^x)) ), x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    m = 20;
CoefficientList[(x+1)^((x+1)^(((x+1)^(x+1))^(x+1))) + O[x]^(m+1), x]* Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

Formula

E.g.f: (x+1)^((x+1)^(((x+1)^(x+1))^(x+1))).

A215691 n-th derivative of (x^x)^(x^(x^x)) at x=1.

Original entry on oeis.org

1, 1, 4, 18, 124, 950, 8688, 89600, 1038392, 13309272, 186471480, 2837173152, 46466835072, 815532508440, 15246845864040, 302533865599800, 6344720827608384, 140208886623418752, 3254819745378435264, 79172189409906466560, 2013138139856523598080
Offset: 0

Views

Author

Alois P. Heinz, Aug 20 2012

Keywords

Comments

Also n-th derivative of (x^(x^(x^x)))^x = x^(x^(x^x)*x) at x=1.

Links

Crossrefs

Column k=13 of A215703.
Cf. A005727, A179230, A179405, A215522, A215524, A215643.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^(x^(x^x))), x, n+1), x, n):
seq(a(n), n=0..30);
  • Mathematica
    m = 20;
CoefficientList[(x+1)^((x+1)^((x+1)^(x+1)+1)) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

Formula

E.g.f.: (x+1)^((x+1)^((x+1)^(x+1)+1)).

A215705 n-th derivative of x^((x^x)^x) at x=1.

Original entry on oeis.org

1, 1, 2, 15, 80, 590, 5034, 47110, 511216, 6019416, 77899320, 1092871824, 16459538952, 265695302808, 4560878625744, 83020743848760, 1595943389477760, 32291354360340672, 685838983512807360, 15248888357184824256, 354130117874225585280, 8571971677758345319680
Offset: 0

Views

Author

Alois P. Heinz, Aug 21 2012

Keywords

Comments

Also n-th derivative of x^(x^(x^2)) at x=1.
First term < 0: a(272).

Links

Crossrefs

Column k=7 of A215703.
Cf. A005727, A179230, A179405, A215524.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, x^((x^x)^x) ), x, n+1), x, n):
seq(a(n), n=0..25);
  • Mathematica
    With[{nn=30},CoefficientList[Series[(x+1)^((x+1)^((x+1)^2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 30 2013 *)

Formula

E.g.f.: (x+1)^((x+1)^((x+1)^2)).

A215706 n-th derivative of (((x^x)^x)^x)^x at x=1.

Original entry on oeis.org

1, 1, 8, 48, 344, 3160, 31776, 349440, 4270304, 56343456, 794577600, 11975388480, 191431339392, 3225851451264, 57152333898240, 1061030230525440, 20569247105571840, 415385999498849280, 8719401647417757696, 189836589049809334272, 4278839631584572661760
Offset: 0

Views

Author

Alois P. Heinz, Aug 21 2012

Keywords

Comments

Also n-th derivative of x^(x^4) at x=1.
First term < 0: a(130).

Links

Crossrefs

Column k=9 of A215703.
Cf. A005727, A179230, A179405, A215524.

Programs

  • Maple
    a:= n-> n!*coeff(series(subs(x=x+1, (((x^x)^x)^x)^x ), x, n+1), x, n):
seq(a(n), n=0..25);

Formula

E.g.f.: (x+1)^((x+1)^4).
Showing 1-10 of 15 results. Next

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

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