Search: id:A008970
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%I A008970
%S A008970 1,1,2,1,6,5,1,14,29,16,1,30,118,150,61,1,62,418,926,841,272,1,126,
%T A008970 1383,4788,7311,5166,1385,1,254,4407,22548,51663,59982,34649,7936,1,
%U A008970 510,13736,100530,325446,553410,517496,252750,50521,1,1022,42236
%N A008970 Triangle T(n,k) = P(n,k)/2, n >= 2, 1<=k<n, of one-half of number of 
               permutations of 1..n such that the differences have k runs with the 
               same signs.
%D A008970 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261, #13, P_{n,k}.
%D A008970 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 260, Table 7.2.1.
%H A008970 M. Bona and R. Ehrenborg, <a href="http://arXiv.org/abs/math.CO/9902020">
               [math/9902020] A combinatorial proof of the log-concavity of the 
               numbers of permutations with k runs</a>
%F A008970 Let P(n, k) = number of permutations of [1..n] with k "sequences". Note 
               that A008970 gives P(n, k)/2. Then g.f.: Sum_{n, k} P(n, k)*u^k*t^n/
               n! = (1+u)^(-1)*((1-u)*(1-sin(v+t*cos(v))-1) where u = sin v.
%F A008970 P(n, 1)=2, P(n, k) = k*P(n-1, k) + 2*P(n-1, k-1) + (n-k)*P(n-1, k-2).
%e A008970 1; 1,2; 1,6,5; 1,14,29,16; ...
%Y A008970 Diagonals give A000352, A000486, A000506, A000111, A000708, A091303. 
               A059427 gives triangle of P(n, k).
%Y A008970 Sequence in context: A145960 A108767 A046817 this_sequence A055896 A159965 
               A116395
%Y A008970 Adjacent sequences: A008967 A008968 A008969 this_sequence A008971 A008972 
               A008973
%K A008970 tabl,nonn,easy,nice
%O A008970 2,3
%A A008970 N. J. A. Sloane (njas(AT)research.att.com).
%E A008970 More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001

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