A196244 -id:A196244 - cp's OEIS frontend

A199205 Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

1, 1, 2, 4, 9, 17, 30, 50, 77, 113, 156, 212, 279, 355, 447, 560, 684, 822, 985, 1171, 1375, 1599, 1856, 2134, 2445, 2769, 3125, 3519, 3939, 4376, 4857, 5372, 5914, 6484, 7083, 7717, 8411, 9130, 9882, 10683, 11524, 12393

Offset: 1

Alois P. Heinz, Nov 03 2011

nonn

			a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215834. Column k=4 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[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]],
                 h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[3], f(n))):
seq(a(n), n=1..20);

f[n_] := f[n] = If[n == 1, {{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]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

a(41)-a(42) from Alois P. Heinz, Jun 01 2015

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.

Original entry on oeis.org

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);

cp's OEIS Frontend

A199205 Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple