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 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. 13
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 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 FORMULA A(n,k) = 1/n * [(d/dx)^n x^^k]_{x=1}. A(n,k) = (n-1)! * [x^n] (x+1)^^k. A(n,k) = Sum_{i=0..min(n,k)} A295027(n,i). A(n,k) = 1/n * A277537(n,k). EXAMPLE 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, ... 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 *) CROSSREFS Columns k=1-10 give: A063524, A005168, A295103, A295104, A295105, A295106, A295107, A295108, A295109, A295110. Main diagonal gives A136461(n-1). Cf. A277537, A295027. Sequence in context: A293202 A280265 A292795 * A294201 A079618 A151844 Adjacent sequences:  A295025 A295026 A295027 * A295029 A295030 A295031 KEYWORD sign,tabl AUTHOR Alois P. Heinz, Nov 12 2017 STATUS approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)