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%I #22 Mar 08 2020 00:51:52
%S 1,1,1,2,1,8,2,1,22,28,2,1,52,182,72,2,1,114,864,974,164,2,1,240,3474,
%T 8444,4174,352,2,1,494,12660,57194,61464,15782,732,2,1,1004,43358,
%U 332528,660842,373940,55286,1496,2,1,2026,142552,1747558,5814124
%N Triangle of number of permutations of [n] with 0 successions, by number of rises.
%C The recurrence given by Roselle is wrong.
%H D. P. Roselle, <a href="https://dx.doi.org/10.1090/S0002-9939-1968-0218256-9">Permutations by number of rises and successions</a>, Proc. Amer. Math. Soc., 19 (1968), 8-16.
%H D. P. Roselle, <a href="/A046739/a046739.pdf">Permutations by number of rises and successions</a>, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
%F a(n, 1) = 1; for r > 1, a(n, r) = r*a(n-1, r) + (n-r)*a(n-1, r-1) + (n-2)*a(n-2, r-1).
%F a(n, 2) = 2^n - 2*n = 2*A000295 = A005803, n >= 3.
%e Triangle begins:
%e 1;
%e 1;
%e 1, 2;
%e 1, 8, 2;
%e 1, 22, 28, 2;
%e ...
%t a[_, 1] = 1; a[n_, 2] := 2^n - 2*n; a[n_, r_] /; 1 <= r <= n-1 := a[n, r] = r*a[n-1, r] + (n-r)*a[n-1, r-1] + (n-2)*a[n-2, r-1]; a[_, _] = 0;
%t row[1] = {{1}}; row[n_] := Table[a[n, r], {r, 1, n-1}];
%t Table[row[n], {n, 1, 11}] // Flatten (* _Jean-François Alcover_, Sep 07 2017 *)
%Y Cf. A046739, A000295. Row sums give A000255. Diagonals give A005803, A065340.
%Y Row sums give A000255.
%K nonn,easy,nice,tabf
%O 1,4
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Jan 03 2003
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