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

1, 1, 3, 7, 49, 201, 1411, 7183, 108417, 816049, 9966691, 80843511, 1381416433, 14049020857, 216003063459, 2309595457471, 72927332784001, 1046829280528353, 23403341433961027, 329565129021010279, 9695176730057249841, 160632514329660035881

Offset: 0

Author

Paul D. Hanna, May 07 2006

Keywords

• nonn

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

Crossrefs

Cf. A059344, variants: A118395, A118930.

Programs

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

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 *)

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

Formula

a(n) = Sum_{k=0..[n/2]} n!/k!/(n-2*k)! *a(k) for n>=0, with a(0)=1.