A263831 -id:A263831 - cp's OEIS frontend

A211192 Consider all distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways; sequence gives difference between numbers of f with f(0)=1 and numbers of f with f(0)=0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

0, -1, 1, 0, 2, 1, 8, 10, 39, 72, 225, 506, 1434, 3550, 9767, 25391, 69293, 185061, 505843, 1372744, 3769842, 10339104, 28546539, 78890525, 218945822, 608657861, 1697106780, 4740593393, 13272626627, 37224982494, 104599603493, 294384019508, 829836855332

Offset: 0

Alois P. Heinz, Feb 18 2013

sign

A000081(n) distinct functions are representable as x -> x^x^...^x with n 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(n-1) for n>0.

			There are A000081(4) = 4 functions f representable as x -> x^x^...^x with 4 x's and parentheses inserted in all possible ways: ((x^x)^x)^x, (x^x)^(x^x) == (x^(x^x))^x, x^((x^x)^x), x^(x^(x^x)).  Only x^((x^x)^x) evaluates to 0 at x=0: 0^((0^0)^0) = 0^(1^0) = 0^1 = 0.  Three functions evaluate to 1 at x=0: ((0^0)^0)^0 = (1^0)^0 = 1^0 = 1, (0^0)^(0^0) = 1^1 = 1, 0^(0^(0^0)) = 0^(0^1) = 0^0 = 1. Thus a(4) = 3-1 = 2.
a(8) = A222380(8) - A222379(8) = 77 - 38 = 39.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, Plot of A000081(8) = 115 = 77 + 38 functions with 8 x's
Wikipedia, Zero to the power of zero

Cf. A000081, A000108, A055113, A215703, A222379, A222380, A258592, A263831, A306668.

g:= proc(n, i) option remember; `if`(n=0, [0, 1], `if`(i<1, 0, (v->[v[1]-
      v[2], v[2]])(add(((l, h)-> [binomial(l[2]+l[1]+j-1, j)*(h[1]+h[2]),
      binomial(l[1]+j-1, j)*h[2]])(g(i-1$2), g(n-i*j, i-1)), j=0..n/i))))
    end:
a:= n-> (f-> f[1]-f[2])(g(n-1$2)):
seq(a(n), n=0..40);

g[n_, i_] := g[n, i] = If[n==0, {0, 1}, If[i<1, {0, 0}, ({#[[1]]-#[[2]], #[[2]]}&)[Sum[Function[{l, h}, {(h[[1]]+h[[2]])*Binomial[j+l[[1]]+l[[2]] -1, j], h[[2]]*Binomial[j+l[[1]]-1, j]}][g[i-1, i-1]], g[n-i*j, i-1]]], {j, 0, Quotient[n, i]}]];
a[n_] := (#[[1]]-#[[2]]&)[g[n-1, n-1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

a(n) = A222380(n) - A222379(n).
From Alois P. Heinz, Mar 01 2019: (Start)
a(n) is even <=> n in { A258592 }.
a(n) is odd  <=> n in { A263831 }. (End)

A263832 The number c_{Cc,pi_1(B_2)}(n) of the second amphicosm n-coverings over the second amphicosm.

Original entry on oeis.org

1, 0, 5, 2, 7, 0, 9, 6, 18, 0, 13, 10, 15, 0, 35, 14, 19, 0, 21, 14, 45, 0, 25, 30, 38, 0, 58, 18, 31, 0, 33, 30, 65, 0, 63, 36, 39, 0, 75, 42, 43, 0, 45, 26, 126, 0, 49, 70, 66, 0, 95, 30, 55, 0, 91, 54, 105, 0, 61, 70, 63, 0, 162, 62, 105, 0, 69

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..20000
G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
G. Chelnokov, M. Deryagina and A. Mednykh, On the coverings of Euclidean manifolds B_1 and B_2, Communications in Algebra, Vol. 45, No. 4 (2017), 1558-1576.

Cf. A263825, A263826, A263827, A263828, A263829, A263830, A263831.

sigma[n_] := DivisorSigma[1, n]; q = Quotient;
a[n_] := Switch[Mod[n, 4], 0, Sum[sigma[q[n, 2d]] - sigma[q[n, 4d]], {d, Divisors[q[n, 4]]}], 2, 0, 1|3, Sum[sigma[d], {d, Divisors[n]}]];
Array[a, 70] (* Jean-François Alcover, Dec 01 2018, after Gheorghe Coserea *)

A007429(n) = sumdiv(n, d, sigma(d));
a(n) = {
  if (n%2, A007429(n), if (n%4, 0,
      sumdiv(n\4, d, sigma(n\(2*d)) - sigma(n\(4*d)))));
};
vector(67, n, a(n))  \\ Gheorghe Coserea, May 05 2016

More terms from Gheorghe Coserea, May 05 2016

A258592 Values of k such that the number of rooted trees with k nodes (A000081(k)) is even.

Original entry on oeis.org

0, 3, 4, 6, 7, 9, 11, 12, 13, 19, 20, 21, 24, 26, 29, 31, 32, 34, 36, 37, 39, 41, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 66, 69, 70, 72, 79, 80, 81, 83, 85, 86, 88, 89, 90, 91, 92, 94, 95, 96, 97, 100, 101, 102, 103, 105, 106, 107

Offset: 1

Vladimir Reshetnikov, Nov 06 2015

nonn

Complement of A263831.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000081, A211192, A263831.

Module[{t}, t[1] = 1; t[k_] := t[k] = Sum[DivisorSum[k-m, t[#] # &] t[m]/(k-1), {m, k-1}]; Select[Range[0, 107], EvenQ@t[#] &]] (* after Alois P. Heinz *)

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A211192 Consider all distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways; sequence gives difference between numbers of f with f(0)=1 and numbers of f with f(0)=0, with conventions that 0^0=1^0=1^1=1, 0^1=0.

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A263832 The number c_{Cc,pi_1(B_2)}(n) of the second amphicosm n-coverings over the second amphicosm.

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A258592 Values of k such that the number of rooted trees with k nodes (A000081(k)) is even.

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