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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. 9
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)).

REFERENCES

J. Alman, C. Lian, B. Tran, Circular Planar Electrical Networks: Posets and Positivity, 2013; http://www.math.umn.edu/~reiner/REU/AlmanLianTran2013.pdf

LINKS

Table of n, a(n) for n=0..18.

FORMULA

G.f.: A(x) = x/series_reversion(x*G(x)); G(x) = A(x*G(x)); A(x) = G(x/A(x)); where G(x) is the g.f. of the double factorials (A001147). - Paul D. Hanna, Jul 09 2006

EXAMPLE

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.

A(x) = 1 + x + 2*x^2 + 8*x^3 + 52*x^4 + 464*x^5 + 5184*x^6 +... = x/series_reversion(x + x^2 + 3*x^3 + 15*x^4 + 105*x^5 + 945*x^6 +...).

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 *)

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

CROSSREFS

Cf. A001147, A232967.

Sequence in context: A136794 A125787 A007832 * A006351 A089467 A195192

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

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Last modified April 25 03:36 EDT 2014. Contains 240994 sequences.