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%I A340766 #35 May 29 2022 13:36:30
%S A340766 1,3,7,17,43,106,273,678,1759,4389,11430,28614,74685,187433,489926,
%T A340766 1231957,3223387,8118434,21256897,53609282,140442534,354595210,
%U A340766 929326086,2348710733,6157476873,15575365846,40843347873,103392210473,271181242774,686944588009
%N A340766 Number of ordered subsequences of {1,...,2n} containing at least n elements and such that the first differences contain only odd numbers.
%H A340766 Alois P. Heinz, <a href="/A340766/b340766.txt">Table of n, a(n) for n = 0..2413</a>
%F A340766 a(n) = A345123(2n,n).
%F A340766 a(n) ~ c * (27/4)^(n/2) / sqrt(3*Pi*n/2), where c = 14 if n is even and c = 8*sqrt(3) if n is odd. Equivalently, c = 7 + 4*sqrt(3) + (7 - 4*sqrt(3))*(-1)^n. - _Vaclav Kotesovec_, Jun 19 2021
%e A340766 a(3) = 17: [1,2,3], [1,2,5], [1,4,5], [2,3,4], [2,3,6], [2,5,6], [3,4,5], [4,5,6], [1,2,3,4], [1,2,3,6], [1,2,5,6], [1,4,5,6], [2,3,4,5], [3,4,5,6], [1,2,3,4,5], [2,3,4,5,6], [1,2,3,4,5,6].
%p A340766 g:= proc(n, k) option remember; `if`(k>n, 0,
%p A340766      `if`(k in [0, 1], n^k, g(n-1, k-1)+g(n-2, k)))
%p A340766     end:
%p A340766 b:= proc(n, k) option remember;
%p A340766      `if`(k>n, 0, g(n, k)+b(n, k+1))
%p A340766     end:
%p A340766 a:= n-> b(2*n, n):
%p A340766 seq(a(n), n=0..30);
%t A340766 g[n_, k_] := g[n, k] = Which[k > n, 0, k == 0, 1, k == 1, n,
%t A340766      True, g[n - 1, k - 1] + g[n - 2, k]];
%t A340766 b[n_, k_] := b[n, k] = If[k > n, 0, g[n, k] + b[n, k + 1]];
%t A340766 a[n_] := b[2*n, n];
%t A340766 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 29 2022, after _Alois P. Heinz_ *)
%Y A340766 Cf. A345123.
%K A340766 nonn
%O A340766 0,2
%A A340766 _Alois P. Heinz_, Jun 10 2021

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