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A118393
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Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ).
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2
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1, 1, 3, 7, 49, 201, 1411, 7183, 108417, 816049, 9966691, 80843511, 1381416433, 14049020857, 216003063459, 2309595457471, 72927332784001, 1046829280528353, 23403341433961027, 329565129021010279, 9695176730057249841, 160632514329660035881
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OFFSET
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0,3
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COMMENTS
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E.g.f. of A059344 is: exp(x+y*x^2). More generally, given a triangle with e.g.f.: exp(x+y*x^b), the eigenvector will have e.g.f.: exp( Sum_{n>=0} x^(b^n) ).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.
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MAPLE
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option remember;
if n <=1 then
1;
else
n!*add(procname(k)/k!/(n-2*k)!, k=0..n/2) ;
end if;
end proc:
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
a(n-j)*binomial(n-1, j-1))(2^i), i=0..ilog2(n)))
end:
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MATHEMATICA
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a[0] = 1; a[n_] := a[n] = Sum[n!/k!/(n - 2*k)!*a[k], {k, 0, n/2}];
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PROG
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(PARI) a(n)=n!*polcoeff(exp(sum(k=0, #binary(n), x^(2^k))+x*O(x^n)), n)
(Sage)
f=factorial;
def a(n): return 1 if n==0 else sum((f(n)/(f(k)*f(n-2*k)))*a(k) for k in (0..n//2))
(Magma)
function a(n)
if n eq 0 then return 1;
else return (&+[ (Factorial(n)/(Factorial(k)*Factorial(n-2*k)))*a(k): k in [0..Floor(n/2)]]);
end if; return a; end function;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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