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A156361 a(2*n+2) = 7*a(2*n+1), a(2*n+1) = 7*a(2*n) - 6^n*A000108(n), a(0) = 1.

%I A156361 #14 Nov 09 2022 19:17:49
%S A156361 1,6,42,288,2016,14040,98280,686880,4808160,33638976,235472832,
%T A156361 1647983232,11535882624,80745019776,565215138432,3956385876480,
%U A156361 27694701135360,193860506096640,1357023542676480,9499115800977408
%N A156361 a(2*n+2) = 7*a(2*n+1), a(2*n+1) = 7*a(2*n) - 6^n*A000108(n), a(0) = 1.
%C A156361 Hankel transform is 6^C(n+1, 2).
%H A156361 G. C. Greubel, <a href="/A156361/b156361.txt">Table of n, a(n) for n = 0..1000</a>
%H A156361 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry2/barry73.html">A Note on a One-Parameter Family of Catalan-Like Numbers</a>, JIS 12 (2009) 09.5.4.
%F A156361 a(n) = Sum{k=0..n} A120730(n,k) * 6^k.
%F A156361 (n+1)*a(n) = 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3). - _R. J. Mathar_, Jul 21 2016
%p A156361 A156361 := proc(n)
%p A156361     option remember;
%p A156361     local nh;
%p A156361     if n= 0 then
%p A156361         1;
%p A156361     elif  type(n,'even') then
%p A156361         7*procname(n-1);
%p A156361     else
%p A156361         nh := floor(n/2) ;
%p A156361         7*procname(n-1)-6^nh*A000108(nh) ;
%p A156361     end if;
%p A156361 end proc: # _R. J. Mathar_, Jul 21 2016
%t A156361 a[n_]:= a[n]= If[n==0, 1, 7*a[n-1] -If[EvenQ[n], 0, 6^((n-1)/2)* CatalanNumber[(n-1)/2]]];
%t A156361 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 04 2022 *)
%o A156361 (Magma) [n le 3 select Factorial(n+4)/120 else (7*n*Self(n-1) + 24*(n-3)*Self(n-2) - 168*(n-3)*Self(n-3))/n: n in [1..30]]; // _G. C. Greubel_, Nov 09 2022
%o A156361 (SageMath)
%o A156361 def a(n): # a = A156361
%o A156361     if (n==0): return 1
%o A156361     elif (n%2==1): return 7*a(n-1) - 6^((n-1)/2)*catalan_number((n-1)/2)
%o A156361     else: return 7*a(n-1)
%o A156361 [a(n) for n in (0..30)] # _G. C. Greubel_, Nov 09 2022
%Y A156361 Cf. A000108, A001405, A156128, A151162, A156195, A151254, A151281.
%K A156361 nonn
%O A156361 0,2
%A A156361 _Philippe Deléham_, Feb 08 2009