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A138413 A bisection of A000957.

%I A138413 #16 Apr 26 2023 18:29:50
%S A138413 0,0,2,18,186,2120,25724,325878,4260282,57048048,778483932,
%T A138413 10786724388,151355847012,2146336125648,30711521221376,
%U A138413 442862000693438,6429286894263738,93891870710425440,1378379704593824300,20330047491994213884,301111732041234778316,4476705468260134734384,66784808491631598524136
%N A138413 A bisection of A000957.
%F A138413 Conjecture: D-finite with recurrence 4*n*(2*n-1)*(7*n-13)*a(n) +(-910*n^3+3489*n^2-4277*n+1680)*a(n-1) +2*(4*n-7)*(7*n-6)*(4*n-5)*a(n-2)=0. Telescoping would provide another recurrence for A000957. - _R. J. Mathar_, Jun 26 2020
%p A138413 b:= proc(n) option remember; `if`(n<3, n*(2-n),
%p A138413       ((7*n-12)*b(n-1)+(4*n-6)*b(n-2))/(2*n))
%p A138413     end:
%p A138413 a:= n-> b(2*n):
%p A138413 seq(a(n), n=0..25);  # _Alois P. Heinz_, Apr 26 2023
%o A138413 (Python)
%o A138413 from itertools import count, islice
%o A138413 def A138413_gen(): # generator of terms
%o A138413     yield from (0,0)
%o A138413     a, c = 0, 1
%o A138413     for n in count(1,2):
%o A138413         a = (c:=c*((n<<2)+2)//(n+2))-a>>1
%o A138413         yield (a:=(c:=c*((n+1<<2)+2)//(n+3))-a>>1)
%o A138413 A138413_list = list(islice(A138413_gen(),20)) # _Chai Wah Wu_, Apr 26 2023
%Y A138413 Cf. A000957, A138414.
%K A138413 nonn,easy
%O A138413 0,3
%A A138413 _N. J. A. Sloane_, May 08 2008