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 A277537 A(n,k) is 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>=0, k>=0, read by antidiagonals. 14
 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 3, 0, 0, 1, 1, 2, 9, 8, 0, 0, 1, 1, 2, 9, 32, 10, 0, 0, 1, 1, 2, 9, 56, 180, 54, 0, 0, 1, 1, 2, 9, 56, 360, 954, -42, 0, 0, 1, 1, 2, 9, 56, 480, 2934, 6524, 944, 0, 0, 1, 1, 2, 9, 56, 480, 4374, 26054, 45016, -5112, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened Eric Weisstein's World of Mathematics, Power Tower Wikipedia, Knuth's up-arrow notation Wikipedia, Tetration FORMULA A(n,k) = [(d/dx)^n x^^k]_{x=1}. E.g.f. of column k: (x+1)^^k. A(n,k) = Sum_{i=0..min(n,k)} A277536(n,i). A(n,k) = n * A295028(n,k) for n,k > 0. EXAMPLE Square array A(n,k) begins:   1, 1,   1,    1,     1,     1,     1,     1, ...   0, 1,   1,    1,     1,     1,     1,     1, ...   0, 0,   2,    2,     2,     2,     2,     2, ...   0, 0,   3,    9,     9,     9,     9,     9, ...   0, 0,   8,   32,    56,    56,    56,    56, ...   0, 0,  10,  180,   360,   480,   480,   480, ...   0, 0,  54,  954,  2934,  4374,  5094,  5094, ...   0, 0, -42, 6524, 26054, 47894, 60494, 65534, ... MAPLE f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end: A:= (n, k)-> n!*coeff(series(f(k), x, n+1), x, n): seq(seq(A(n, d-n), n=0..d), d=0..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)): seq(seq(A(n, d-n), n=0..d), d=0..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]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 14 2017, adapted from 2nd Maple prog. *) CROSSREFS Columns k=0..10 give A000007, A019590(n+1), A005727, A179230, A179405, A179505, A211205, A277538, A277539, A277540, A277541. Rows n=0..1 give A000012, A057427. Main diagonal gives A033917. Cf. A215703, A277536, A295028. Sequence in context: A035208 A025881 A039804 * A323179 A320508 A164925 Adjacent sequences:  A277534 A277535 A277536 * A277538 A277539 A277540 KEYWORD sign,tabl AUTHOR Alois P. Heinz, Oct 19 2016 STATUS approved

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Last modified February 25 22:44 EST 2020. Contains 332270 sequences. (Running on oeis4.)