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A338526 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] without consecutive adjacent values.

%I A338526 #46 Aug 16 2024 12:02:23
%S A338526 1,1,1,1,2,0,1,3,2,0,1,4,6,4,2,1,5,12,18,20,14,1,6,20,48,90,124,90,1,
%T A338526 7,30,100,272,582,860,646,1,8,42,180,650,1928,4386,6748,5242,1,9,56,
%U A338526 294,1332,5110,15912,37566,59612,47622,1,10,72,448,2450,11604,46250,148648,360642,586540,479306
%N A338526 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] without consecutive adjacent values.
%C A338526 Also number of ways to arrange n non-attacking kings on an n X k board, with 0 or 1 in each row and 1 in each column. - _Ron L.J. van den Burg_, Aug 04 2024
%F A338526 T(n,k) = (n! + Sum_{p=1..k-1} (-1)^p (n-p)! Sum_{r=1..p} 2^r binomial(k-p,r) binomial(p-1,r-1) )/(n-k)!. - _Ron L.J. van den Burg_, Aug 04 2024
%F A338526 O.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k = Sum_{i>=0} i!(x*y*(1-x*y)/(1+x*y))^i/(1-x)^(i+1). - _Ron L.J. van den Burg_, Aug 14 2024
%e A338526 n\k  0    1    2    3    4    5    6    7    8
%e A338526 0    1
%e A338526 1    1    1
%e A338526 2    1    2    0
%e A338526 3    1    3    2    0
%e A338526 4    1    4    6    4    2
%e A338526 5    1    5    12   18   20   14
%e A338526 6    1    6    20   48   90   124  90
%e A338526 7    1    7    30   100  272  582  860  646
%e A338526 8    1    8    42   180  650  1928 4386 6748 5242
%o A338526 (PARI) isok(s, p) = {for (i=1, #s-1, if (abs(s[p[i+1]] - s[p[i]]) == 1, return (0));); return (1);}
%o A338526 T(n, k) = {my(nb = 0); forsubset([n, k], s, for(i=1, k!, if (isok(s, numtoperm(k, i)), nb++););); nb;} \\ _Michel Marcus_, Nov 17 2020
%Y A338526 Diagonal is A002464.
%Y A338526 T(2n,n) gives A375022.
%K A338526 nonn,tabl
%O A338526 0,5
%A A338526 _Xiangyu Chen_, Nov 07 2020