OFFSET
0,3
COMMENTS
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) ).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450
FORMULA
a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.
MAPLE
A118393 := proc(n)
option remember;
if n <=1 then
1;
else
n!*add(procname(k)/k!/(n-2*k)!, k=0..n/2) ;
end if;
end proc:
seq(A118393(n), n=0..20) ; # R. J. Mathar, Aug 19 2014
# 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:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 01 2017
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[n!/k!/(n - 2*k)!*a[k], {k, 0, n/2}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 18 2018 *)
PROG
(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))
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021
(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;
[a(n): n in [0..25]]; // G. C. Greubel, Feb 18 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 07 2006
STATUS
approved