A156361 - OEIS

%I #14 Nov 09 2022 19:17:49

%S 1,6,42,288,2016,14040,98280,686880,4808160,33638976,235472832,

%T 1647983232,11535882624,80745019776,565215138432,3956385876480,

%U 27694701135360,193860506096640,1357023542676480,9499115800977408

%N 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 Hankel transform is 6^C(n+1, 2).

%H G. C. Greubel, <a href="/A156361/b156361.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 a(n) = Sum{k=0..n} A120730(n,k) * 6^k.

%F (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 := proc(n)

%p     option remember;

%p     local nh;

%p     if n= 0 then

%p         1;

%p     elif  type(n,'even') then

%p         7*procname(n-1);

%p     else

%p         nh := floor(n/2) ;

%p         7*procname(n-1)-6^nh*A000108(nh) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Jul 21 2016

%t 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 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 04 2022 *)

%o (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 (SageMath)

%o def a(n): # a = A156361

%o     if (n==0): return 1

%o     elif (n%2==1): return 7*a(n-1) - 6^((n-1)/2)*catalan_number((n-1)/2)

%o     else: return 7*a(n-1)

%o [a(n) for n in (0..30)] # _G. C. Greubel_, Nov 09 2022

%Y Cf. A000108, A001405, A156128, A151162, A156195, A151254, A151281.

%K nonn

%O 0,2

%A _Philippe Deléham_, Feb 08 2009