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 A352448 Expansion of e.g.f. LambertW( -2*x/(1-x) ) / (-2*x). 8
 1, 3, 22, 278, 5128, 125592, 3850000, 142013328, 6129705088, 303238991744, 16920975718144, 1051612647426816, 72045481821580288, 5394849460316820480, 438392509692455286784, 38424395486908104071168, 3613476161122656804438016 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..344 FORMULA E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies: (1) A(x) = LambertW( -2*x/(1-x) ) / (-2*x). (2) A(x) = exp( 2*x*A(x) ) / (1-x). (3) A(x) = log( (1-x) * A(x) ) / (2*x). (4) A( x/(exp(2*x) + x) ) = exp(2*x) + x. (5) A(x) = (1/x) * Series_Reversion( x/(exp(2*x) + x) ). (6) Sum_{k=0..n} [x^k] 1/A(x)^n = 0, for n > 1. (7) [x^(n+1)/(n+1)!] 1/A(x)^n = -2^(n+1) * n for n >= (-1). a(n) ~ (1 + 2*exp(1))^(n + 3/2) * n^(n-1) / (2^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 18 2022 a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Mar 03 2023 EXAMPLE E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 278*x^3/3! + 5128*x^4/4! + 125592*x^5/5! + 3850000*x^6/6! + 142013328*x^7/7! + ... such that A(x) = exp( 2*x*A(x) ) / (1-x), where exp( 2*x*A(x) ) = 1 + 2*x + 16*x^2/2! + 212*x^3/3! + 4016*x^4/4! + 99952*x^5/5! + 3096448*x^6/6! + 115063328*x^7/7! + ... Related table. Another interesting property of the e.g.f. A(x) is illustrated here. The table of coefficients of x^k/k! in 1/A(x)^n begins: n=1: [1, -3, -4, -44, -736, -16832, -491168, ...]; n=2: [1, -6, 10, -16, -320, -8064, -249344, ...]; n=3: [1, -9, 42, -78, -48, -1776, -66528, ...]; n=4: [1, -12, 92, -392, 728, -128, -8960, ...]; n=5: [1, -15, 160, -1120, 4600, -8520, -320, ...]; n=6: [1, -18, 246, -2424, 16104, -64752, 119952, ...]; ... from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1, as follows: n=1:-2 = 1 + -3; n=2: 0 = 1 + -6 + 10/2!; n=3: 0 = 1 + -9 + 42/2! + -78/3!; n=4: 0 = 1 + -12 + 92/2! + -392/3! + 728/4!; n=5: 0 = 1 + -15 + 160/2! + -1120/3! + 4600/4! + -8520/5!; n=6: 0 = 1 + -18 + 246/2! + -2424/3! + 16104/4! + -64752/5! + 119952/6!; ... PROG (PARI) {a(n) = n!*polcoeff( (1/x)*serreverse( x/(exp(2*x +x^2*O(x^n)) + x) ), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) my(x='x+O('x^30)); Vec(serlaplace(lambertw(-2*x/(1-x))/(-2*x))) \\ Michel Marcus, Mar 17 2022 (PARI) a(n) = n!*sum(k=0, n, 2^k*(k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Mar 03 2023 CROSSREFS Cf. A352410, A352411, A352412. Cf. A361068. Sequence in context: A319147 A074706 A293989 * A141360 A162659 A360596 Adjacent sequences: A352445 A352446 A352447 * A352449 A352450 A352451 KEYWORD nonn AUTHOR Paul D. Hanna, Mar 16 2022 STATUS approved

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Last modified August 9 11:22 EDT 2024. Contains 375042 sequences. (Running on oeis4.)