This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A182962 E.g.f. satisfies: A(x) = exp( x/(1 - x*A'(x)/A(x)) ). 19
 1, 1, 3, 25, 433, 12501, 529531, 30495613, 2272643745, 211761416233, 24055076979091, 3267213865097601, 522451410607362193, 97120159467079471165, 20765771676360919883403, 5060640084128464622069221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..250 FORMULA E.g.f.: A(x) = exp(x*F(x)) where F(x) = 1 + x*F(x)*d/dx[x*F(x)] is the o.g.f. of A088716. E.g.f. satisfies: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n for n>=1. E.g.f. satisfies: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n) for n>=0. E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where A(x*G(x)) = G(x) is the e.g.f. of A156326, which satisfies: . G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ). a(n) ~ c * (n!)^2 * n, where c = 0.21795078944715106549... (see A238223). - Vaclav Kotesovec, Feb 22 2014 EXAMPLE E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 433*x^4/4! +... The logarithm of the e.g.f. is the integer series: log(A(x)) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 +...+ A088716(n)*x^(n+1) +... ... The coefficients of [x^n/n!] in the powers of e.g.f. A(x) begin: A^1: [(1),(1), 3, 25, 433, 12501, 529531, 30495613, ...]; A^2: [1,(2),(8), 68, 1120, 30832, 1260544, 70737536, ...]; A^3: [1, 3,(15),(135), 2169, 57303, 2261439, 123523515, ...]; A^4: [1, 4, 24,(232),(3712), 94944, 3622336, 192461056, ...]; A^5: [1, 5, 35, 365, (5905),(147625), 5460475, 282185825, ...]; A^6: [1, 6, 48, 540, 8928, (220176),(7926336), 398625408, ...]; A^7: [1, 7, 63, 763, 12985, 318507,(11210479),(549313471), ...]; A^8: [1, 8, 80, 1040, 18304, 449728, 15551104,(743759360), ...]; ... In the above table, the coefficients in parenthesis are related by: 1*1 = 1; 8 = 2^2*2; 135 = 3^2*15; 3712 = 4^2*232; 147625 = 5^2*5905; this illustrates: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n. ... Also note that the main diagonal in the above table begins: [1*1, 2*1, 3*5, 4*58, 5*1181, 6*36696, 7*1601497, 8*92969920, ...]; this illustrates: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n). ... Let G(x) denote the e.g.f. of A156326: G(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +... then G(x) satisfies: G(x) = A(x*G(x)) and A(x) = G(x/A(x)) where G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ). ... MATHEMATICA m = 16; A[_] = 1; Do[A[x_] = Exp[x/(1 - x A'[x]/A[x])] + O[x]^m, {m}]; CoefficientList[A[x], x] Range[0, m-1]! (* Jean-François Alcover, Oct 29 2019 *) PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x/(1 - x*deriv(A)/A+x*O(x^n)))); n!*polcoeff(A, n)} (PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=((#A-1)*Vec(Ser(A)^(#A-1))[#A-1]-Vec(Ser(A)^(#A-1))[#A])/(#A-1)); n!*A[n+1]} CROSSREFS Cf. A088716, A300735, A300986, A300988, A300990, A300992. Cf. A156326, A238223. Sequence in context: A323217 A160143 A009843 * A223076 A272482 A136173 Adjacent sequences:  A182959 A182960 A182961 * A182963 A182964 A182965 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 01 2011 STATUS approved

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

Last modified December 14 14:20 EST 2019. Contains 329979 sequences. (Running on oeis4.)