A003019 -id:A003019 - cp's OEIS frontend

A199296 Number of distinct values taken by 5th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

1, 1, 2, 4, 9, 20, 45, 92, 182, 342, 601, 982, 1499, 2169, 2970, 3994, 5297, 6834, 8635, 10714, 13121, 16104, 19674, 23868, 28453, 33637, 39630, 46730

Offset: 1

Alois P. Heinz, Nov 04 2011

nonn

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 5th derivative at x=1: (x^(x^(x^x))) -> 360; (x^((x^x)^x)) -> 590; ((x^(x^x))^x), ((x^x)^(x^x)) -> 650; (((x^x)^x)^x) -> 1110.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215835. Column k=5 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0, 0]},
            {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
             8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
             10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
             +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4]],
              h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[4], f(n))):
seq(a(n), n=1..20);

f[n_] := f[n] = If[n == 1, {{0, 0, 0, 0}}, Union@Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]], 10 + 50*h[[1]] + 10*h[[2]] + 5*h[[3]] + (30 + 30*h[[1]] + 10*h[[2]] + 15*g[[1]])*g[[1]] + (20 + 10*h[[1]])*g[[2]] + 5*g[[3]] + g[[4]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length@Union@(#[[4]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)

A199883 Number of distinct values taken by 6th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 113, 262, 591, 1263, 2505, 4764, 8479, 14285, 22871, 35316, 52755, 76517, 107826, 148914, 202715, 270622

Offset: 1

Alois P. Heinz, Nov 11 2011

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 6th derivative at x=1: (x^(x^(x^x))) -> 2934; ((x^x)^(x^x)), ((x^(x^x))^x) -> 4908; (x^((x^x)^x)) -> 5034; (((x^x)^x)^x) -> 8322.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215836. Column k=6 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0, 0, 0]},
                {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
                 8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
                 10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
                 +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4],
                 45*h[1]*g[1]^2+(120+60*h[2]+15*h[3]+60*g[2]+
                 270*h[1])*g[1]+54+15*h[3]+30*g[3]+6*g[4]+
                 60*h[1]*g[2]+15*h[1]*g[3]+30*h[1]+ 20*h[2]*g[2]+
                 100*h[2]+90*h[1]^2+g[5]+60*g[2]+6*h[4]],
                 h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[5], f(n))):
seq(a(n), n=1..15);

a(22)-a(23) from Alois P. Heinz, Sep 26 2014

A215796 Number of distinct values taken by 7th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 283, 691, 1681, 3988, 9241, 20681, 44217, 89644

Offset: 1

Alois P. Heinz, Aug 24 2012

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 7th derivative at x=1: (x^(x^(x^x))) -> 26054; ((x^x)^(x^x)), ((x^(x^x))^x) -> 41090; (x^((x^x)^x)) -> 47110; (((x^x)^x)^x) -> 70098.

Column k=7 of A216368.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215837.

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-> nops({map(f-> 7!*coeff(series(subs(x=x+1, f), x, 8), x, 7), T(n))[]}):
seq(a(n), n=1..12);

A215971 Number of distinct values taken by 8th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 717, 1815, 4574, 11505, 28546, 69705, 166010

Offset: 1

Alois P. Heinz, Aug 29 2012

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 8th derivative at x=1: (x^(x^(x^x))) -> 269128; ((x^x)^(x^x)), ((x^(x^x))^x) -> 382520; (x^((x^x)^x)) -> 511216; (((x^x)^x)^x) -> 646272.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215838. Column k=8 of A216368.

# load programs from A215703, then:
a:= n-> nops({map(f-> 8!*coeff(series(subs(x=x+1, f),
                  x, 9), x, 8), T(n))[]}):
seq(a(n), n=1..10);

A216062 Number of distinct values taken by 9th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1838, 4734, 12247, 31617, 81208

Offset: 1

Alois P. Heinz, Aug 31 2012

a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 9th derivative at x=1: (x^(x^(x^x))) -> 3010680; ((x^x)^(x^x)), ((x^(x^x))^x) -> 3863808; (x^((x^x)^x)) -> 6019416; (((x^x)^x)^x) -> 6333336.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215839. Column k=9 of A216368.

# load programs from A215703, then:
a:= n-> nops({map(f-> 9!*coeff(series(subs(x=x+1, f),
                  x, 10), x, 9), T(n))[]}):
seq(a(n), n=1..11);

A216403 Number of distinct values taken by 10th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4763, 12452, 32711, 86239

Offset: 1

Alois P. Heinz, Sep 06 2012

nonn

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 10th derivative at x=1: (x^(x^(x^x))) -> 37616880; ((x^x)^(x^x)), ((x^(x^x))^x) -> 42409440; (x^((x^x)^x)) -> 77899320; (((x^x)^x)^x) -> 66712680.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215840. Column k=10 of A216368.

# load programs from A215703, then:
a:= n-> nops({map(f-> 10!*coeff(series(subs(x=x+1, f),
                  x, 11), x, 10), T(n))[]}):
seq(a(n), n=1..11);

A082543 Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible ways. Sequence gives number of distinct integer values when x=sqrt(2).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 5, 7

Offset: 1

Benoit Cloitre, May 02 2003

It appears that the integer values obtained are always of form 2^(2^k).

Cf. A003019, A082499.

More terms and comments from W. Edwin Clark, May 02 2003

cp's OEIS Frontend

A199296 Number of distinct values taken by 5th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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Mathematica

A199883 Number of distinct values taken by 6th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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A215796 Number of distinct values taken by 7th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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Maple

A215971 Number of distinct values taken by 8th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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Maple

A216062 Number of distinct values taken by 9th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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Maple

A216403 Number of distinct values taken by 10th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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Maple

A082543 Take a string of n x's and insert n-1 ^'s and n-1 pairs of parentheses in all possible ways. Sequence gives number of distinct integer values when x=sqrt(2).

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