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A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

%I A118396 #9 Nov 07 2020 05:34:45
%S A118396 1,1,1,7,25,61,481,2731,10417,454105,4309921,23452111,592433161,
%T A118396 6789801877,46254009985,893881991731,11548704851041,93501748795441,
%U A118396 4828847934591937,83867376656907415,823025819684123641,33409213329178701421,640457721676922946721
%N A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).
%C A118396 E.g.f. of triangle A118394 is: exp(x+y*x^3), where A118394(n,k) = n!/k!/(n-3*k)!. 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 A118396 Alois P. Heinz, <a href="/A118396/b118396.txt">Table of n, a(n) for n = 0..450</a>
%F A118396 a(n) = Sum_{k=0..[n/3]} n!/k!/(n-3*k)! *a(k) for n>=0, with a(0)=1.
%p A118396 a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
%p A118396       a(n-j)*binomial(n-1, j-1))(3^i), i=0..ilog[3](n)))
%p A118396     end:
%p A118396 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 01 2017
%t A118396 a[n_] := a[n] = If[n==0, 1, Sum[Function[j, j! a[n-j] Binomial[n-1, j-1]][3^i], {i, 0, Log[3, n]}]];
%t A118396 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 07 2020, after _Alois P. Heinz_ *)
%o A118396 (PARI) {a(n)=n!*polcoeff(exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k))+x*O(x^n)),n)}
%Y A118396 Cf. A118394, A118395; variants: A118393, A118932.
%K A118396 nonn
%O A118396 0,4
%A A118396 _Paul D. Hanna_, May 07 2006