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 A352410 Expansion of e.g.f. LambertW( -x/(1-x) ) / (-x). 17
 1, 2, 9, 67, 717, 10141, 179353, 3816989, 95076537, 2714895433, 87457961421, 3138260371225, 124147801973605, 5368353187693757, 251928853285058433, 12752446755011776741, 692625349011401620209, 40178978855796929378065, 2479383850197948228950293 (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..370 FORMULA E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies: (1) A(x) = LambertW( -x/(1-x) ) / (-x). (2) A(x) = exp( x*A(x) ) / (1-x). (3) A(x) = log( (1-x) * A(x) ) / x. (4) A( x/(exp(x) + x) ) = exp(x) + x. (5) A(x) = (1/x) * Series_Reversion( x/(exp(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 = -n for n >= (-1). a(n) ~ (1 + exp(1))^(n + 3/2) * n^(n-1) / exp(n + 1/2). - Vaclav Kotesovec, Mar 15 2022 a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Sep 24 2022 EXAMPLE E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 67*x^3/3! + 717*x^4/4! + 10141*x^5/5! + 179353*x^6/6! + 3816989*x^7/7! + ... such that A(x) = exp(x*A(x)) / (1-x), where exp(x*A(x)) = 1 + x + 5*x^2/2! + 40*x^3/3! + 449*x^4/4! + 6556*x^5/5! + 118507*x^6/6! + ... + A052868(n)*x^n/n! + ... which equals LambertW(-x/(1-x)) * (1-x)/(-x). Related table. Another defining 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, -2, -1, -7, -71, -961, -16409, -339571, ...]; n=2: [1, -4, 6, -2, -24, -362, -6644, -144538, ...]; n=3: [1, -6, 21, -33, -3, -63, -1395, -34275, ...]; n=4: [1, -8, 44, -148, 232, -4, -152, -4876, ...]; n=5: [1, -10, 75, -395, 1305, -2045, -5, -355, ...]; n=6: [1, -12, 114, -822, 4224, -13806, 21636, -6, ...]; n=7: [1, -14, 161, -1477, 10381, -52507, 170401, -267043, -7, ...]; ... 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:-1 = 1 + -2; n=2: 0 = 1 + -4 + 6/2!; n=3: 0 = 1 + -6 + 21/2! + -33/3!; n=4: 0 = 1 + -8 + 44/2! + -148/3! + 232/4!; n=5: 0 = 1 + -10 + 75/2! + -395/3! + 1305/4! + -2045/5!; n=6: 0 = 1 + -12 + 114/2! + -822/3! + 4224/4! + -13806/5! + 21636/6!; n=7: 0 = 1 + -14 + 161/2! + -1477/3! + 10381/4! + -52507/5! + 170401/6! + -267043/7!; ... PROG (PARI) {a(n) = n!*polcoeff( (1/x)*serreverse( x/(exp(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(-x/(1-x))/(-x))) \\ Michel Marcus, Mar 17 2022 (PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Sep 24 2022 CROSSREFS Cf. A352411, A352412, A352448, A052868. Cf. A102743, A108919, A331726. Sequence in context: A324167 A296793 A366336 * A336588 A322612 A364337 Adjacent sequences: A352407 A352408 A352409 * A352411 A352412 A352413 KEYWORD nonn AUTHOR Paul D. Hanna, Mar 15 2022 STATUS approved

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Last modified August 11 12:11 EDT 2024. Contains 375069 sequences. (Running on oeis4.)