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!)
 A111088 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 2. 10
 1, 1, 2, 8, 52, 464, 5184, 68928, 1057584, 18345536, 354570112, 7551674624, 175700025728, 4433961734656, 120642462777344, 3520972469815296, 109731998026937088, 3637456413350962176, 127800512612435896320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For x = 1, this is : 1, 1, 1, 2, 7, 34, 206, 1476, 12123, ..., see A075834. For x = 0, this is : 1, 1, 0, 0, 0, 0, 0, 0, 0, ... For x = -1, this is : 1, 1, -1, 2, -5, 14, -42, 132, -429, ...,((-1)^(n+1)* A000108(n)). a(n)*2^(-n) is the coefficient at the x^(n-1) term in the series reversal of the asymptotic expansion of 2 * DawsonF(sqrt(x))/sqrt(x) = sqrt(Pi) * exp(-x) * erfi(sqrt(x)) / sqrt(x) for x -> inf. - Vladimir Reshetnikov, Apr 23 2016 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 J. Alman, C. Lian, B. Tran, Circular Planar Electrical Networks: Posets and Positivity, 2013. FORMULA O.g.f. A(x) satisfies: (1) A(x) = x / Series_Reversion(x*G(x)) where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) such that G(x) is the g.f. of the double factorials (A001147). - Paul D. Hanna, Jul 09 2006 (2) A(x) = Sum_{n>=0} A001147(n) * x^n / A(x)^n, where A001147(n) = (2*n)!/(n!*2^n). - Paul D. Hanna, Aug 02 2014 (3) A(x) = 1 + x * (A(x) + x*A'(x)) / (A(x) - x*A'(x)). - Paul D. Hanna, Aug 02 2014 (4) [x^(n+1)] A(x)^n = 2*n*([x^n] A(x)^n) for n>=0. - Paul D. Hanna, Aug 02 2014 a(n) ~ 2^(n+1/2) * n^n / exp(n+1/2). - Vaclav Kotesovec, Aug 02 2014 EXAMPLE From Paul D. Hanna, Aug 02 2014: (Start) O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 52*x^4 + 464*x^5 + 5184*x^6 +... where A(x) = x/Series_Reversion(x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 945*x^6 +...) and thus A(x) = 1 + x/A(x) + 3*x^2/A(x)^2 + 15*x^3/A(x)^3 + 105*x^4/A(x)^4 + 945*x^5/A(x)^5 +... Illustration of the initial terms: a(2) = 2; a(3) = 2*2^2 = 8; a(4) = 2*3*8 + 1*2*2 = 52; a(5) = 2*4*52 + 1*2*8 + 2*8*2 = 464; a(6) = 2*5*464 + 1*2*52 + 2*8*8 + 3*52*2 = 5184; ... To illustrate formula: [x^(n+1)] A(x)^n = 2*n*([x^n] A(x)^n), form a table of coefficients of x^k in A(x)^n: n=1: [1, 1,  2,   8,   52,   464,  5184,   68928,  1057584, ...]; n=2: [1, 2,  5,  20,  124,  1064, 11568,  150912,  2283888, ...]; n=3: [1, 3,  9,  37,  222,  1836, 19412,  248256,  3703536, ...]; n=4: [1, 4, 14,  60,  353,  2824, 29032,  363696,  5345040, ...]; n=5: [1, 5, 20,  90,  525,  4081, 40810,  500480,  7241460, ...]; n=6: [1, 6, 27, 128,  747,  5670, 55205,  662460,  9431172, ...]; n=7: [1, 7, 35, 175, 1029,  7665, 72765,  854197, 11958758, ...]; n=8: [1, 8, 44, 232, 1382, 10152, 94140, 1081080, 14876033, ...]; ... then we can see that the diagonals are related in the following way: [2, 20, 222, 2824, 40810, 662460, 11958758, ...] = [2*1, 4*5, 6*37, 8*353, 10*4081, 12*55205, 14*854197, ...]. Also, the diagonal [1, 5, 37, 353, 4081, 55205, 854197, 14876033, ...] is the logarithmic derivative of the g.f. of the double factorials. Further, the main diagonal in the above table equals: [1, 2*1, 3*3, 4*15, 5*105, 6*945, 7*10395, 8*135135, ...]. (End) MATHEMATICA x = 2; a[0] = a[1] = 1; a[2] = x; a[3] = 2x^2; a[n_] := a[n] = x*(n - 1)*a[n - 1] + Sum[(j - 1)*a[j]*a[n - j], {j, 2, n - 2}]; Table[ a[n], {n, 0, 18}] (* Robert G. Wilson v *) Module[{max = 20, s}, s = InverseSeries[Series[2 DawsonF[Sqrt[x]]/Sqrt[x], {x, Infinity, max + 1}][[2, 2, 2]]]; Table[SeriesCoefficient[s, n-1] 2^n, {n, 0, max}]] (* Vladimir Reshetnikov, Apr 23 2016 *) PROG (PARI) a(n)=Vec(x/serreverse(x*Ser(vector(n+1, k, (2*(k-1))!/(k-1)!/2^(k-1)))))[n+1] /* Paul D. Hanna, Jul 09 2006 */ (PARI) /* From o.g.f. A = 1 + x*(A + x*A')/(A - x*A'): */ {a(n)=local(A=1+x); for(i=1, n, A=1 + x*(A+x*A')/(A-x*A' +x*O(x^n))); polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) /* Paul D. Hanna, Aug 02 2014 */ CROSSREFS Cf. A001147, A232967, A000699. Sequence in context: A136794 A125787 A007832 * A006351 A300697 A277499 Adjacent sequences:  A111085 A111086 A111087 * A111089 A111090 A111091 KEYWORD nonn AUTHOR Philippe Deléham, Oct 10 2005 EXTENSIONS More terms from Robert G. Wilson v, Oct 12 2005 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 5 16:21 EST 2019. Contains 329753 sequences. (Running on oeis4.)