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a(0)=1, otherwise a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*a(n-1-2*k).
5

%I #24 Jun 04 2022 09:42:09

%S 1,1,1,2,4,8,19,46,118,322,903,2653,8053,25194,81387,269667,917529,

%T 3197480,11393821,41497060,154186653,584151512,2254240317,8852998343,

%U 35361762709,143540660088,591802631729,2476701062087

%N a(0)=1, otherwise a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*a(n-1-2*k).

%H Seiichi Manyama, <a href="/A172383/b172383.txt">Table of n, a(n) for n = 0..932</a>

%F G.f. A(x) satisfies: A(x) = 1 + (x/(1-x^2)) * A(x/(1-x^2)).

%e Eigensequence for number triangle

%e 1;

%e 1, 0;

%e 0, 1, 0;

%e 1, 0, 1, 0;

%e 0, 2, 0, 1, 0;

%e 1, 0, 3, 0, 1, 0;

%e 0, 3, 0, 4, 0, 1, 0;

%e 1, 0, 6, 0, 5, 0, 1, 0;

%e 0, 4, 0, 10, 0, 6, 0, 1, 0;

%e 1, 0, 10, 0, 15, 0, 7, 0, 1, 0;

%e 0, 5, 0, 20, 0, 21, 0, 8, 0, 1, 0;

%e (augmented version of Riordan array (1/(1-x^2), x/(1-x^2)), A030528.

%p A172383 := proc(n)

%p option remember;

%p if n = 0 then

%p 1;

%p else

%p add(binomial(n-k-1,k)*procname(n-1-2*k),k=0..floor((n-1)/2)) ;

%p end if;

%p end proc:

%p seq(A172383(n),n=0..20) ; # _R. J. Mathar_, Feb 11 2015

%t a[n_]:= If[n == 0, 1, Sum[Binomial[n-k-1, k]*a[n-2*k-1], {k, 0, Floor[(n-1)/2]}]]; Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Oct 07 2018 *)

%Y Cf. A030528.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Feb 01 2010

%E Name corrected by _R. J. Mathar_, Feb 11 2015