login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

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..1000

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 A277499 A089467

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 4 16:52 EST 2016. Contains 278750 sequences.