The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A306091 G.f. A(x) satisfies: (1 + A(x))^A(x) = (1 + x)^x ; this sequence gives the denominators of the coefficients of x^n in g.f. A(x). 3
 1, 2, 4, 6, 8, 1440, 960, 120960, 48384, 7257600, 1612800, 479001600, 4561920, 5230697472000, 10461394944000, 7846046208000, 6974263296000, 9146248151040000, 8536498274304000, 1502674769756160000, 1857852442607616000, 67440043666656460800000, 44960029111104307200000, 18613452051997183180800000, 954536002666522214400000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The numerators of the coefficients in g.f. A(x) are given by A306090. LINKS Paul D. Hanna, Table of n, a(n) for n = 1..300 FORMULA G.f. A(x) = Sum_{n>=0} A306090(n)/A306091(n) * x^n satisfies: (1) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + k*A(x) = 1. (2) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + (k - p)*A(x) = (1 + x)^p. (3) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + k*A(x) = (1 + A(x))^m. (4) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + (k - p)*A(x) = (1+x)^p * (1 + A(x))^m. (5) A(A(x)) = x. (6) (1 + A(x))^A(x) = (1 + x)^x. (7) Sum_{n>=1} (-A(x))^(n+1) / n = x*log(1+x). (8) Let F(x,y) = Series_Reversion( (exp(-x*y) - exp(-x))/(1-y) ), where the inverse is taken wrt x, and let F'(x,y) = d/dx F(x,y), then F'(x, A(x)/x) = 1 (derived from Peter Bala's g.f. for A067948). EXAMPLE G.f.: A(x) = -x + 1/2*x^2 - 1/4*x^3 + 1/6*x^4 - 1/8*x^5 + 143/1440*x^6 - 79/960*x^7 + 8483/120960*x^8 - 2953/48384*x^9 + 391753/7257600*x^10 - 77983/1612800*x^11 + 20963473/479001600*x^12 - 182269/4561920*x^13 + 192178874539/5230697472000*x^14 - 355629691849/10461394944000*x^15 + 248105704337/7846046208000*x^16 - 206101262483/6974263296000*x^17 + 253628381647657/9146248151040000*x^18 - 222936799599583/8536498274304000*x^19 + 37078279922025269/1502674769756160000*x^20 + ... + A306090(n)/A306091(n)*x^n + ... such that (E.1) 1 = 1 + (x + A(x)) + (x + 2*A(x))*(2*x + A(x))/2! + (x + 3*A(x))*(2*x + 2*A(x))*(3*x + A(x))/3! + (x + 4*A(x))*(2*x + 3*A(x))*(3*x + 2*A(x))*(4*x + A(x))/4! + (x + 5*A(x))*(2*x + 4*A(x))*(3*x + 3*A(x))*(4*x + 2*A(x))*(5*x + A(x))/5! + ... (E.2) (1 + x)^p = 1 + (x + (1-p)*A(x)) + (x + (2-p)*A(x))*(2*x + (1-p)*A(x))/2! + (x + (3-p)*A(x))*(2*x + (2-p)*A(x))*(3*x + (1-p)*A(x))/3! + (x + (4-p)*A(x))*(2*x + (3-p)*A(x))*(3*x + (2-p)*A(x))*(4*x + (1-p)*A(x))/4! + ... (E.3) (1 + A(x))^m = 1 + ((1-m)*x + A(x)) + ((1-m)*x + 2*A(x))*((2-m)*x + A(x))/2! + ((1-m)*x + 3*A(x))*((2-m)*x + 2*A(x))*((3-m)*x + A(x))/3! + ((1-m)*x + 4*A(x))*((2-m)*x + 3*A(x))*((3-m)*x + 2*A(x))*((4-m)*x + A(x))/4! + ... FUNCTIONAL EQUATIONS. The series A(x) satisfies: (E.4) (1 + A(x))^A(x) = (1 + x)^x = 1 + x^2 - 1/2*x^3 + 5/6*x^4 - 3/4*x^5 + 33/40*x^6 - 5/6*x^7 + 2159/2520*x^8 - 209/240*x^9 + ... GENERATING METHOD. Although the functional equation (1 + A(x))^A(x) = (1 + x)^x has an infinite number of solutions, one may arrive at the g.f. A(x) by the following iteration. If we start with A = -x, and iterate (E.5) A = (A + x*log(1 + x)/log(1 + A))/2 then A will converge to g.f. A(x). MATHEMATICA nmax = 25; sol = {a[1] -> -1}; Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[(1 + A[x])^A[x] - (1 + x)^x + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax + 1}]; sol /. Rule -> Set; a /@ Range[1, nmax] // Denominator (* Jean-François Alcover, Nov 02 2019 *) PROG (PARI) /* From Functional Equation (1 + A(x))^A(x) = (1 + x)^x */ {a(n) = my(A = -x +x*O(x^n)); for(i=1, n, A = (A + x*log(1+x +x*O(x^n))/log(1+A))/2 ); denominator( polcoeff(A, n) )} CROSSREFS Cf. A306090 (numerators). Sequence in context: A351777 A030149 A083146 * A074108 A231408 A361973 Adjacent sequences: A306088 A306089 A306090 * A306092 A306093 A306094 KEYWORD nonn,frac AUTHOR Paul D. Hanna, Jun 21 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 21 13:19 EDT 2024. Contains 372736 sequences. (Running on oeis4.)