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 A355378 Expansion of e.g.f. exp(exp(3*x) - exp(x)). 6
 1, 2, 12, 82, 688, 6754, 75096, 928386, 12591392, 185384130, 2938319144, 49799613538, 897495547184, 17118975292514, 344206910941624, 7270287035936706, 160826794265399360, 3716047107259486082, 89472755268582494792, 2240097688067896960674, 58207872357772581544272 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..467 FORMULA a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k, -1). a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 30 2022 a(n) ~ exp(exp(3*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) - (1 + z)*exp(z)), where z = LambertW(n)/3 - 1/(1 + 3/LambertW(n) - 9 * n^(2/3) * (1 + LambertW(n)) / LambertW(n)^(5/3)). - Vaclav Kotesovec, Jul 03 2022 a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) - (n/LambertW(n))^(1/3) - n) / sqrt(1 + LambertW(n)). - Vaclav Kotesovec, Jul 10 2022 MATHEMATICA nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]! Table[Sum[Binomial[n, k] * 3^k * BellB[k] * BellB[n-k, -1], {k, 0, n}], {n, 0, 20}] PROG (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) - exp(x)))) \\ Michel Marcus, Jun 30 2022 CROSSREFS Cf. A143405, A355291, A355379, A355381. Sequence in context: A199420 A235348 A052864 * A136278 A130464 A319326 Adjacent sequences: A355375 A355376 A355377 * A355379 A355380 A355381 KEYWORD nonn AUTHOR Vaclav Kotesovec, Jun 30 2022 STATUS approved

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Last modified February 21 05:17 EST 2024. Contains 370219 sequences. (Running on oeis4.)