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A118393 Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ).

%I A118393 #16 Feb 19 2021 09:46:04
%S A118393 1,1,3,7,49,201,1411,7183,108417,816049,9966691,80843511,1381416433,
%T A118393 14049020857,216003063459,2309595457471,72927332784001,
%U A118393 1046829280528353,23403341433961027,329565129021010279,9695176730057249841,160632514329660035881
%N A118393 Eigenvector of triangle A059344. E.g.f.: exp( Sum_{n>=0} x^(2^n) ).
%C A118393 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) ).
%H A118393 Alois P. Heinz, <a href="/A118393/b118393.txt">Table of n, a(n) for n = 0..450</a>
%F A118393 a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.
%p A118393 A118393 := proc(n)
%p A118393     option remember;
%p A118393     if n <=1 then
%p A118393         1;
%p A118393     else
%p A118393         n!*add(procname(k)/k!/(n-2*k)!,k=0..n/2) ;
%p A118393     end if;
%p A118393 end proc:
%p A118393 seq(A118393(n),n=0..20) ; # _R. J. Mathar_, Aug 19 2014
%p A118393 # second Maple program:
%p A118393 a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
%p A118393       a(n-j)*binomial(n-1, j-1))(2^i), i=0..ilog2(n)))
%p A118393     end:
%p A118393 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 01 2017
%t A118393 a[0] = 1; a[n_] := a[n] = Sum[n!/k!/(n - 2*k)!*a[k], {k, 0, n/2}];
%t A118393 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 18 2018 *)
%o A118393 (PARI) a(n)=n!*polcoeff(exp(sum(k=0,#binary(n),x^(2^k))+x*O(x^n)),n)
%o A118393 (Sage)
%o A118393 f=factorial;
%o A118393 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))
%o A118393 [a(n) for n in (0..25)] # _G. C. Greubel_, Feb 18 2021
%o A118393 (Magma)
%o A118393 function a(n)
%o A118393   if n eq 0 then return 1;
%o A118393   else return (&+[ (Factorial(n)/(Factorial(k)*Factorial(n-2*k)))*a(k): k in [0..Floor(n/2)]]);
%o A118393   end if; return a; end function;
%o A118393 [a(n): n in [0..25]]; // _G. C. Greubel_, Feb 18 2021
%Y A118393 Cf. A059344, variants: A118395, A118930.
%K A118393 nonn
%O A118393 0,3
%A A118393 _Paul D. Hanna_, May 07 2006