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A005264
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Number of labeled rooted Greg trees with n nodes.
(Formerly M3096)
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13
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1, 3, 22, 262, 4336, 91984, 2381408, 72800928, 2566606784, 102515201984, 4575271116032, 225649908491264, 12187240730230528, 715392567595403520, 45349581052869924352, 3087516727770990992896, 224691760916830871873536
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OFFSET
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1,2
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COMMENTS
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A rooted Greg tree can be described as a rooted tree with 2-colored nodes where only the black nodes are counted and labeled and the white nodes have at least 2 children. - Christian G. Bower, Nov 15 1999
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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D. J. Jeffrey, G. A. Kalugin, and N. Murdoch, Lagrange inversion and Lambert W, Preprint, 2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC).
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FORMULA
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The REVEGF transform of the odd numbers [1,3,5,7,9,11,...] is [1, -3, 22, -262, 4336, -91984, 2381408, ...] - N. J. A. Sloane, May 26 2017
E.g.f. A(x) = y satisfies y' = (1 + 2*y) / ((1 - 2*y) * (1 + x)). - Michael Somos, Mar 26 2011
E.g.f. A(x) satisfies (1 + x) * exp(A(x)) = 1 + 2 * A(x).
A(x) satisfies the separable differential equation A'(x) = exp(A(x))/(1-2*A(x)) with A(0) = 0. Thus the inverse function A^-1(x) = int {t = 0..x} (1-2*t)/exp(t) = exp(-x)*(2*x+1)-1 = x-3*x^2/2!+5*x^3/3!-7*x^4/4!+.... A(x) = -1/2-LambertW(-exp(-1/2)*(x+1)/2).
The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx(exp(x)/(1-2*x)*g(x)). Compare with [Dominici, Example 9].
(End)
a(n)=sum(k=1..n-1, (n+k-1)!*sum(j=1..k, 1/(k-j)!*sum(l=1..j, 1/(l!*(j-l)!)* sum(i=0..n+j-1, ((-1)^(i+l)*l^i*binomial(l,n+j-i-1)*2^(n+j-i-1))/i!)))), n>1, a(1)=1. - Vladimir Kruchinin, May 04 2012
Let T(n,k) = 1 if k=0 and (n=0 or n=1); T(n,k) = 0 if k<0 or k>n; and otherwise T(n,k) = (n-1)*T(n-1,k-1)+(3*n-k-4)*T(n-1,k)-(k+1)*T(n-1,k+1). Define polynomials p(n,w) = w^n*sum_{k=0..n-1}(T(n,k)*w^k)/(1+w)^(2*n-1), then a(n) = (-1)^n*p(n,-1/2). - Peter Luschny, Nov 10 2012
a(n) ~ n^(n-1) / (sqrt(2) * exp(n/2) * (2-exp(1/2))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013
E.g.f.: -W(-(1+x)*exp(-1/2)/2)-1/2 where W is the Lambert W function. - Robert Israel, Mar 28 2017
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EXAMPLE
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G.f. = x + 3*x^2 + 22*x^3 + 262*x^4 + 4336*x^5 + 91984*x^6 + 2381408*x^7 + ...
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MAPLE
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T := proc(n, k) option remember; if k=0 and (n=0 or n=1) then return(1) fi; if k<0 or k>n then return(0) fi;
(n-1)*T(n-1, k-1)+(3*n-k-4)*T(n-1, k)-(k+1)*T(n-1, k+1) end:
A005264 := proc(n) add(T(n, k)*(-1)^k*2^(n-k-1), k=0..n-1) end;
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MATHEMATICA
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max = 17; f[x_] := -1/2 - ProductLog[-E^(-1/2)*(x + 1)/2]; Rest[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!] (* Jean-François Alcover, May 23 2012, after Peter Bala *)
a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ InverseSeries[ Series[ Exp[-x] (1 + 2 x) - 1, {x, 0, n}]], n]]; (* Michael Somos, Jun 07 2012 *)
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PROG
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(PARI) {a(n) = local(A); if( n<1, 0, for( k= 1, n, A += x * O(x^k); A = truncate( (1 + x) * exp(A) - 1 - A) ); n! * polcoeff( A, n))}; /* Michael Somos, Apr 02 2007 */
(PARI) {a(n) = if( n<1, 0, n! * polcoeff( serreverse( exp( -x + x * O(x^n) ) * (1 + 2*x) - 1), n))}; /* Michael Somos, Mar 26 2011 */
(Maxima) a(n):=if n=1 then 1 else sum((n+k-1)!*sum(1/(k-j)!*sum(1/(l!*(j-l)!)*sum(((-1)^(i+l)*l^i*binomial(l, n+j-i-1)*2^(n+j-i-1))/i!, i, 0, n+j-1), l, 1, j), j, 1, k), k, 1, n-1); /* Vladimir Kruchinin, May 04 2012 */
(Sage)
@CachedFunction
def T(n, k):
if k==0 and (n==0 or n==1): return 1
if k<0 or k>n: return 0
return (n-1)*T(n-1, k-1)+(3*n-k-4)*T(n-1, k)-(k+1)*T(n-1, k+1)
A005264 = lambda n: add(T(n, k)*(-1)^k*2^(n-k-1) for k in (0..n-1))
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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