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 A290023 a(n) is the 2n-th derivative of the difference between the n-th tetration of x (power tower of order n) and its predecessor (or 0 if n=0) at x=1. 2
 1, 0, 8, 900, 224112, 78775200, 40518181440, 28340179227360, 26078095792869120, 30544708065077606400, 44428404658605222528000, 78604530683773395984883200, 166295474965751756924207462400, 414658685362517268992110471680000, 1203746810444949373635048911870976000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..100 Eric Weisstein's World of Mathematics, Power Tower Wikipedia, Knuth's up-arrow notation Wikipedia, Tetration FORMULA a(n) = (2n)! * [x^(2n)] (x+1)^^n - (x+1)^^(n-1) for n>0, a(0) = 1. a(n) = [(d/dx)^(2n) (x^^n - x^^(n-1))]_{x=1} for n>0, a(0) = 1. a(n) = A277536(2n,n). MAPLE f:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1, (x+1)^f(n-1))) end: a:= n-> (2*n)!*coeff(series(f(n)-f(n-1), x, 2*n+1), x, 2*n): seq(a(n), n=0..15); # 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-> b(2*n, n) -`if`(n=0, 0, b(2*n, n-1)): seq(a(n), n=0..15); MATHEMATICA f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]]; a[n_] := (2*n)!*SeriesCoefficient[f[n] - f[n - 1], {x, 0, 2*n}]; Table[a[n], {n, 0, 15}] (* Second program: *) 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_] := b[2*n, n] - If[n == 0, 0, b[2*n, n - 1]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 02 2018, from Maple *) CROSSREFS Cf. A277536. Sequence in context: A175411 A027725 A265239 * A112578 A112579 A064072 Adjacent sequences: A290020 A290021 A290022 * A290024 A290025 A290026 KEYWORD nonn AUTHOR Alois P. Heinz, Nov 10 2017 STATUS approved

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Last modified September 15 09:55 EDT 2024. Contains 375932 sequences. (Running on oeis4.)