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A276312 Number of up-down sequences of length n and values in {1,2,...,n}.

%I A276312 #12 Dec 29 2020 09:04:07
%S A276312 1,1,1,5,31,246,2353,26585,345775,5094220,83833256,1524414737,
%T A276312 30353430420,656851828075,15350023574061,385261255931365,
%U A276312 10335781852020335,295166535640444376,8939894824857438940,286234265613041061128,9659753724363828753408
%N A276312 Number of up-down sequences of length n and values in {1,2,...,n}.
%H A276312 Alois P. Heinz, <a href="/A276312/b276312.txt">Table of n, a(n) for n = 0..413</a>
%F A276312 a(n) ~ exp(-1/2) * 2^(n+2) * n^n / Pi^(n+1). - _Vaclav Kotesovec_, Aug 30 2016
%e A276312 a(0) = 1: the empty sequence.
%e A276312 a(1) = 1: 1.
%e A276312 a(2) = 1: 12.
%e A276312 a(3) = 5: 121, 131, 132, 231, 232.
%e A276312 a(4) = 31: 1212, 1213, 1214, 1312, 1313, 1314, 1323, 1324, 1412, 1413, 1414, 1423, 1424, 1434, 2312, 2313, 2314, 2323, 2324, 2412, 2413, 2414, 2423, 2424, 2434, 3412, 3413, 3414, 3423, 3424, 3434.
%p A276312 b:= proc(n, k, t) option remember; `if`(n=0, 1,
%p A276312       add(b(n-1, k, k-j), j=1..t-1))
%p A276312     end:
%p A276312 a:= n-> b(n, n+1$2):
%p A276312 seq(a(n), n=0..25);
%t A276312 b[n_, k_, t_] := b[n, k, t] = If[n==0, 1, Sum[b[n-1, k, k-j], {j, 1, t-1}]];
%t A276312 a[n_] := b[n, n+1, n+1];
%t A276312 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 29 2020, after _Alois P. Heinz_ *)
%Y A276312 A diagonal of A050446, A050447.
%Y A276312 Cf. A276313.
%K A276312 nonn
%O A276312 0,4
%A A276312 _Alois P. Heinz_, Aug 29 2016