A295028 A(n,k) is (1/n) times the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=1, k>=1, read by antidiagonals.
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 8, 2, 0, 1, 1, 3, 14, 36, 9, 0, 1, 1, 3, 14, 72, 159, -6, 0, 1, 1, 3, 14, 96, 489, 932, 118, 0, 1, 1, 3, 14, 96, 729, 3722, 5627, -568, 0, 1, 1, 3, 14, 96, 849, 6842, 33641, 40016, 4716, 0
Offset: 1
Examples
Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 3, 3, 3, 3, 3, 3, ... 0, 2, 8, 14, 14, 14, 14, 14, ... 0, 2, 36, 72, 96, 96, 96, 96, ... 0, 9, 159, 489, 729, 849, 849, 849, ... 0, -6, 932, 3722, 6842, 8642, 9362, 9362, ... 0, 118, 5627, 33641, 71861, 102941, 118061, 123101, ...
Links
- Alois P. Heinz, Antidiagonals n = 1..141, flattened
- Eric Weisstein's World of Mathematics, Power Tower
- Wikipedia, Knuth's up-arrow notation
- Wikipedia, Tetration
Crossrefs
Programs
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Maple
f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end: A:= (n, k)-> (n-1)!*coeff(series(f(k), x, n+1), x, n): seq(seq(A(n, 1+d-n), n=1..d), d=1..14); # 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: A:= (n, k)-> b(n, min(k, n))/n: seq(seq(A(n, 1+d-n), n=1..d), d=1..14); -
Mathematica
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}]]]; A[n_, k_] := b[n, Min[k, n]]/n; Table[A[n, 1 + d - n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 25 2018, translated from 2nd Maple program *)