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A124430 Eigenvector of triangle A124428.

%I A124430 #8 Feb 24 2019 16:31:00
%S A124430 1,1,2,3,7,13,31,61,144,296,714,1534,3761,8303,20495,46115,114461,
%T A124430 261445,651114,1503207,3749017,8726147,21788311,51072555,127698665,
%U A124430 301244477,754496298,1790598079,4494019431,10726676701,26983034009
%N A124430 Eigenvector of triangle A124428.
%H A124430 G. C. Greubel, <a href="/A124430/b124430.txt">Table of n, a(n) for n = 0..1000</a>
%F A124430 a(n) = Sum_{k=0..[n/2]} a(k)*C([n/2],k)*C([(n+1)/2],k) for n>0, with a(0)=1 and [] means floor().
%e A124430 a(5) = 1*a(0) + 6*a(1) + 3*a(2) = 1*1 + 6*1 + 3*2 = 13;
%e A124430 a(6) = 1*a(0) + 9*a(1) + 9*a(2) + 1*a(3) = 1*1 + 9*1 + 9*2 + 1*3 = 31.
%e A124430 Triangle A124428(n,k) = C([n/2],k)*C([(n+1)/2],k) begins:
%e A124430   1;
%e A124430   1;
%e A124430   1,  1;
%e A124430   1,  2;
%e A124430   1,  4,  1;
%e A124430   1,  6,  3;
%e A124430   1,  9,  9,  1;
%e A124430   1, 12, 18,  4;
%e A124430   1, 16, 36, 16,  1; ...
%t A124430 a[n_]:= a[n] = If[n==0, 1, Sum[Binomial[Floor[n/2], k]*Binomial[Floor[(n + 1)/2], k]*a[k], {k,0,Floor[n/2]}]]; Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Feb 24 2019 *)
%o A124430 (PARI) {a(n)=if(n==0,1,sum(k=0,n\2,a(k)*binomial(n\2,k)*binomial((n+1)\2,k)))}
%Y A124430 Cf. A124428, A124429.
%K A124430 nonn
%O A124430 0,3
%A A124430 _Paul D. Hanna_, Oct 31 2006