A033917 -id:A033917 - cp's OEIS frontend

A277529 Decimal expansion of the eighth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

5, 0, 1, 0, 9, 2, 7, 6, 2, 7, 5, 0, 3, 6, 6, 9, 6, 8, 9, 5, 8, 6, 8, 0, 1, 7, 2, 3, 5, 3, 2, 2, 9, 4, 5, 1, 2, 7, 2, 1, 8, 3, 9, 4, 2, 3, 8, 3, 2, 4, 2, 6, 2, 4, 4, 6, 0, 2, 8, 9, 9, 9, 1, 1, 8, 5, 8, 4, 6, 3, 2, 4, 1, 1, 0, 1, 8, 9, 6, 2, 0, 2, 8, 2, 1, 0, 6, 6, 7, 3, 8, 4, 1, 4, 3, 1, 6, 6, 8, 7, 3, 6, 1, 0, 8, 7

Offset: 5

Alois P. Heinz, Oct 19 2016

			-50109.2762750366968958680172353229451272183942...

Alois P. Heinz, Table of n, a(n) for n = 5..1004
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277525, A277526, A277527, A277528, A277530, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[8][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A277530 Decimal expansion of the ninth derivative of the infinite power tower function x^x^x... at x = 1/2.

Original entry on oeis.org

8, 4, 4, 6, 1, 7, 0, 8, 6, 7, 1, 0, 9, 3, 6, 1, 1, 9, 7, 5, 1, 5, 8, 2, 0, 7, 1, 1, 4, 9, 3, 8, 3, 2, 2, 3, 9, 5, 4, 6, 1, 0, 0, 2, 1, 7, 3, 6, 6, 2, 7, 5, 5, 1, 8, 4, 4, 1, 7, 2, 9, 4, 8, 3, 6, 5, 8, 0, 2, 4, 8, 4, 2, 5, 9, 8, 1, 7, 4, 6, 6, 3, 5, 1, 9, 8, 8, 5, 9, 6, 2, 2, 4, 9, 1, 5, 0, 1, 6, 9, 8, 4, 3, 5, 1

Offset: 6

Alois P. Heinz, Oct 19 2016

			844617.08671093611975158207114938322395461002...

Alois P. Heinz, Table of n, a(n) for n = 6..1005
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277525, A277526, A277527, A277528, A277529, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[9][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A277531 Decimal expansion of the tenth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

Original entry on oeis.org

1, 4, 1, 0, 9, 5, 0, 8, 6, 0, 4, 5, 7, 8, 2, 5, 2, 1, 6, 6, 7, 4, 8, 0, 7, 9, 5, 6, 6, 6, 2, 1, 3, 8, 6, 1, 1, 5, 9, 2, 7, 6, 0, 7, 4, 9, 5, 2, 5, 0, 6, 8, 9, 5, 9, 1, 6, 1, 0, 1, 8, 4, 7, 0, 8, 2, 4, 0, 0, 4, 4, 5, 8, 4, 4, 8, 8, 7, 2, 4, 0, 0, 8, 9, 3, 2, 4, 1, 1, 6, 2, 1, 3, 3, 3, 3, 4, 9, 8, 0, 6, 7, 0, 5, 3

Offset: 8

Alois P. Heinz, Oct 19 2016

			-14109508.6045782521667480795666213861159276...

Alois P. Heinz, Table of n, a(n) for n = 8..1007
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277525, A277526, A277527, A277528, A277529, A277530.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[10][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A216349 Triangle T(n,k) in which n-th row lists the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

Original entry on oeis.org

1, 2, 12, 9, 156, 100, 80, 56, 3160, 1880, 1180, 1420, 950, 1360, 890, 660, 480, 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, 14988, 10848, 34974, 21474, 13314, 15114, 10974, 13014, 8874, 6534, 5094, 3218628, 1806476, 1021552, 588756, 1189132

Offset: 1

Alois P. Heinz, Sep 04 2012

The ordering of the functions is the same as in A215703 and is defined by the algorithm below.

			For n=4 the A000081(4) = 4 functions and their 4th derivatives at x=1 are x^(x^3)->156, x^(x^x*x)->100, x^(x^(x^2))->80, x^(x^(x^x))->56.
Triangle T(n,k) begins:
:     1;
:     2;
:    12,     9;
:   156,   100,    80,    56;
:  3160,  1880,  1180,  1420,   950,  1360,   890,   660,   480;
: 87990, 50496, 29682, 35382, 24042, 22008, 14928, 31968, 20268, ...

Alois P. Heinz, Rows n = 1..12, flattened

First column gives: A216351.
Last elements of rows give: A033917.
A version with sorted row elements is: A216350.
Rows sums give: A216281.
Cf. A000081, A215703.

with(combinat):
F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^n],
     `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
      w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
    end:
T:= n-> map(f-> n!*coeff(series(subs(x=x+1, f), x, n+1), x, n), F(n))[]:
seq(T(n), n=1..7);

A216350 Triangle T(n,k) in which n-th row lists in increasing order the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

Original entry on oeis.org

1, 2, 9, 12, 56, 80, 100, 156, 480, 660, 890, 950, 1180, 1360, 1420, 1880, 3160, 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, 20268, 21474, 22008, 24042, 29682, 31968, 34974, 35382, 50496, 87990, 65534, 78134, 102494, 131684, 141974

Offset: 1

Alois P. Heinz, Sep 04 2012

			For n=4 the A000081(4) = 4 functions and their 4th derivatives at x=1 are x^(x^3)->156, x^(x^x*x)->100, x^(x^(x^2))->80, x^(x^(x^x))->56 => 4th row = [56, 80, 100, 156].
Triangle T(n,k) begins:
:    1;
:    2;
:    9,   12;
:   56,   80,  100,   156;
:  480,  660,  890,   950,  1180,  1360,  1420,  1880,  3160;
: 5094, 6534, 8874, 10848, 10974, 13014, 13314, 14928, 14988, 15114, ...

Alois P. Heinz, Rows n = 1..12, flattened

First column gives: A033917.
Last elements of rows give: A216351.
A version with different ordering of row elements is: A216349.
Rows sums give: A216281.
Cf. A000081, A215703.

with(combinat):
F:= proc(n) F(n):= `if`(n<2, [x$n], map(h->x^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [x^n],
     `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
      w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
    end:
T:= n-> sort(map(f-> n!*coeff(series(subs(x=x+1, f)
                 , x, n+1), x, n), F(n)))[]:
seq(T(n), n=1..7);

A349505 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^3).

Original entry on oeis.org

1, 1, 6, 81, 1668, 46740, 1659102, 71386602, 3611376360, 210083758704, 13817649943440, 1013979735381888, 82134894774767832, 7279520816638839600, 700730732176038359208, 72803537907677356262760, 8120227815636769492383168

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..336

Cf. A008275, A033917, A264408, A349504, A349525.

nmax = 20; CoefficientList[Series[(-LambertW[-3*Log[1 + x]]/(3*Log[1 + x]))^(1/3), {x, 0, nmax}], x]*Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)

a(n) = sum(k=0, n, (3*k+1)^(k-1)*stirling(n, k, 1));

N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*log(1+x)^k/k!)))

a(n) = Sum_{k=0..n} (3*k+1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (log(1+x))^k / k!.
From Vaclav Kotesovec, Nov 20 2021: (Start)
E.g.f.: (-LambertW(-3*log(1 + x)) / (3*log(1 + x)))^(1/3).
a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n + exp(-1)/6 - 5/6)). (End)

A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 3, 6, 0, 0, 8, 24, 24, 0, 0, 10, 170, 180, 120, 0, 0, 54, 900, 1980, 1440, 720, 0, 0, -42, 6566, 19530, 21840, 12600, 5040, 0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320, 0, 0, -5112, 365256, 2650536, 4818744, 4536000, 2993760, 1270080, 362880

Offset: 0

Alois P. Heinz, Oct 19 2016

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0,   2;
  0, 0,   3,     6;
  0, 0,   8,    24,     24;
  0, 0,  10,   170,    180,    120;
  0, 0,  54,   900,   1980,   1440,    720;
  0, 0, -42,  6566,  19530,  21840,  12600,   5040;
  0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Columns k=0-2 give: A000007, A063524, A005727 (for n>1).
Main diagonal gives A000142.
Row sums give A033917.
T(n+1,n)/3 gives A005990.
T(2n,n) gives A290023.
Cf. A277537, A295027.

f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
T:= (n, k)-> n!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
      -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
      (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
    end:
T:= (n, k)-> b(n, min(k, n))-`if`(k=0, 0, b(n, min(k-1, n))):
seq(seq(T(n, k), k=0..n), n=0..12);

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := n!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
(* second program: *)
b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0, 0, -Sum[Binomial[n - 1, j]*b[j, k]*Sum[Binomial[n - j, i]*(-1)^i*b[n - j - i, k - 1]*(i - 1)!, {i, 1, n - j}], {j, 0, n - 1}]]];
T[n_, k_] := (b[n, Min[k, n]] - If[k == 0, 0, b[n, Min[k - 1, n]]]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 28 2018, from Maple *)

E.g.f. of column k>0: (x+1)^^k - (x+1)^^(k-1), e.g.f. of column k=0: 1.
T(n,k) = [(d/dx)^n (x^^k - x^^(k-1))]_{x=1} for k>0, T(n,0) = A000007(n).
T(n,k) = A277537(n,k) - A277537(n,k-1) for k>0, T(n,0) = A000007(n).
T(n,k) = n * A295027(n,k) for n,k > 0.

A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).

Original entry on oeis.org

1, 1, 4, 36, 484, 8840, 203868, 5691308, 186612592, 7031373264, 299397454080, 14218443479328, 745142534904480, 42717896158340832, 2659373970144454080, 178666030775042040000, 12884568940594969258752, 992750028716940749121792

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..356

Cf. A008275, A033917, A264407, A349505, A349524.

nmax = 20; CoefficientList[Series[Sqrt[-LambertW[-2*Log[1 + x]]/(2*Log[1 + x])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)

a(n) = sum(k=0,n, (2*k+1)^(k-1)*stirling(n, k, 1));

N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*log(1+x)^k/k!)))

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (log(1+x))^k / k!.
From Vaclav Kotesovec, Nov 20 2021: (Start)
E.g.f.: sqrt(-LambertW(-2*log(1 + x)) / (2*log(1 + x))).
a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1)). (End)

A136461 Expansion of e.g.f.: A(x) = -(1 + LambertW(-log(1+x))/log(1+x))/x.

Original entry on oeis.org

1, 1, 3, 14, 96, 849, 9362, 123101, 1888016, 33066768, 651883152, 14286514186, 344690210928, 9079702374300, 259327537407416, 7983107543564724, 263518937698466304, 9285770278110061664, 347916420499685643072, 13812127364516107258944, 579183295530010157485824

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

A033917 gives the coefficients of iterated exponential function defined by y(x) = x^y(x) expanded about x=1.

Alois P. Heinz, Table of n, a(n) for n = 0..400 (first 71 terms from Vincenzo Librandi)

Cf. A033917.
Row sums of A295027 (shifted).
Main diagonal of A295028 (shifted).

a:= n-> add(Stirling1(n+1, k)*(k+1)^(k-1), k=0..n+1)/(n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2016

CoefficientList[Series[-(1+LambertW[-Log[1+x]]/Log[1+x])/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

{a(n)=n!*polcoeff(sum(i=0,n+1,(i+1)^(i-1)*log(1+x +O(x^(n+2) ))^i/i!), n+1)}

x='x+O('x^30); Vec(serlaplace(-(1+lambertw(-log(1+x))/log(1+x))/x  )) \\ G. C. Greubel, Feb 19 2018

a(n) = A033917(n+1)/(n+1).
E.g.f.: A(x) = (1/x)*Sum_{i>=1} (i+1)^(i-1) * log(1+x)^i/i!.
a(n) ~ n^(n-1) / ( exp(n-3/2+exp(-1)/2) * (exp(exp(-1))-1)^(n+1/2) ). - Vaclav Kotesovec, Nov 27 2012

A277541 n-th derivative of the tenth tetration of x (power tower of order 10) x^^10 at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 65534, 984808, 16992144, 330667680, 7130797872, 168564160632, 4321664793264, 119356965323400, 3528831476247240, 111173720474673984, 3716755785886791168, 131414199676568655552, 4899052003032070987968, 192050612714621129114880

Offset: 0

Alois P. Heinz, Oct 19 2016

nonn

Differs from A033917 first at n=11.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=10 of A277537.

Cf. A033917, A215703, A295110.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> n!*coeff(series(f(10), x, n+1), x, n):
seq(a(n), n=0..25);

f[n_] := f[n] = If[n == 0, 1, (x + 1)^f[n - 1]];
a[n_] := n!*SeriesCoefficient[f[10], {x, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 30 2018, from Maple *)

E.g.f.: (x+1)^^10.

cp's OEIS Frontend

A277529 Decimal expansion of the eighth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

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A277530 Decimal expansion of the ninth derivative of the infinite power tower function x^x^x... at x = 1/2.

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A277531 Decimal expansion of the tenth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

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A216349 Triangle T(n,k) in which n-th row lists the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

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A216350 Triangle T(n,k) in which n-th row lists in increasing order the values of the n-th derivative at x=1 of all functions that are representable as x^x^...^x with n x's and parentheses inserted in all possible ways; n>=1, 1<=k<=A000081(n).

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A349505 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^3).

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A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).

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