OFFSET
0,2
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..350
Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 2016, Volume 172, March 2017, Pages 145-159.
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
FORMULA
G.f. A(x) satisfies (1+x)^2 = A(x/(1+x))^2/A(x). - Michael Somos, Feb 16 2006
G.f.: A(x) = Product_{n>=1} 1/(1 - n*x)^(1/2^n). - Paul D. Hanna, Jun 16 2010
a(n) ~ (n-1)! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 19 2014
From Peter Bala, May 26 2001: (Start)
EXAMPLE
G.f.: A(x) = (1-x)^(-1/2)*(1-2*x)^(-1/4)*(1-3*x)^(-1/8)*(1-4*x)^(-1/16)*... - Paul D. Hanna, Jun 16 2010
MATHEMATICA
nmax = 19; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[(1+x)^2 * A[x] - A[x/(1+x)]^2 + O[x]^(n+1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)
With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[(-2)^j*j!*StirlingS2[k, j], {j, k}]*(-x)^k /k, {k, m+1}]], {x, 0, m}], x]] (* G. C. Greubel, Jun 08 2023 *)
PROG
(PARI) A = matrix(25, 25); A[1, 1] = 1; rs = 1; print(1); for (n=2, 25, sc = sum(i=2, n-1, A[i, 1]*A[n+1-i, 1]); A[n, 1] = rs - sc; rs = A[n, 1]; for (k=2, n, A[n, k] = A[n, k-1] + A[n-1, k-1]; rs += A[n, k]); print(A[n, n])); \\ David Wasserman, Jan 06 2005
(PARI) {a(n)=local(A); if(n<0, 0, A=1; for(k=1, n, A=truncate(A+O(x^k))+x*O(x^k); A+=A-(subst(1/A, x, x/(1+x))*(1+x))^-2; ); polcoeff(A, n))} /* Michael Somos, Feb 18 2006 */
(Magma)
m:=40;
f:= func< n, x | Exp((&+[(&+[(-2)^j*Factorial(j)*StirlingSecond(k, j)*(-x)^k/k: j in [1..k]]): k in [1..n+2]])) >;
R<x>:=PowerSeriesRing(Rationals(), m+1); // A084785
Coefficients(R!( f(m, x) )); // G. C. Greubel, Jun 08 2023
(SageMath)
def f(n, x): return exp(sum(sum( (-2)^j*factorial(j)* stirling_number2(k, j)*(-x)^k/k for j in range(1, k+1)) for k in range(1, n+2)))
m=50
def A084785_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(m, x) ).list()
A084785_list(m-9) # G. C. Greubel, Jun 08 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Jun 13 2003
EXTENSIONS
More terms from David Wasserman, Jan 06 2005
STATUS
approved