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 A326397 Triangle T(n,k) read by rows: T(n,k) = number of ways of seating n people around a table for the second time so that k pairs are maintained. Reflected sequences are counted as one. 4
 1, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 0, 8, 0, 4, 5, 0, 25, 25, 0, 5, 18, 72, 90, 120, 54, 0, 6, 161, 490, 784, 637, 343, 98, 0, 7, 1416, 4352, 5920, 5120, 2416, 768, 160, 0, 8, 13977, 40500, 54027, 42525, 21951, 6723, 1485, 243, 0, 9, 149630, 417400, 535850, 414200, 208100, 70760, 15500, 2600, 350, 0, 10, 1737241, 4691654 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 COMMENTS Definition requires "pairs" and for n=0 it is assumed that there is 1 way of seating 0 people around a table for the second time so that 0 pairs are maintained and 1 person forms only one pair with him/herself. Therefore T(0,0)=1, T(1,0)=0 and T(1,1)=1. Weighted average of each row using k as weights converges to 2 for large n and is given by the following formula: (Sum_{k} T(n,k)*k)/(Sum_{k} T(n,k)) = 2/(n-1) + 2 (conjectured). LINKS Witold Tatkiewicz, Rows n = 0..17 of triangle, flattened Witold Tatkiewicz, link for java program FORMULA T(n,n) = n for n>2. T(n,n-1) = 0 for n>1. T(n,n-3) = 1/2*n^3 + 3/4*n^2 - 2 (conjectured); T(n,n-3) = (2/3)*n^4 + 3*n^3 + (1/3)*n^2 - 7*n + 3 for n > 4 (conjectured); T(n,n-4) = (25/24)*n^5 + (73/12)*n^4 + (5/8)*n^3 - (253/12)*n^2 + (76/3)*n - 12 for n > 5 (conjectured); T(n,n-5) = (26/15)*n^6 + (77/6)*n^5 + 7*n^4 - (97/3)*n^3 + (2314/15)*n^2 - 273/2*n + 65 for n > 5 (conjectured); T(n,n-6) = (707/240)*n^7 + (2093/80)*n^6 + (2009/80)*n^5 - (245/16)*n^4 + (78269/120)*n^3 - (18477/20)*n^2 + (10647/0)*n - 342 for n > 6 (conjectured). EXAMPLE Assuming the initial order was {1,2,3,4,5} (therefore 1 and 5 form a pair as first and last person are neighbors in case of round table) there are 5 sets of ways of seating them again so that 3 pairs are conserved: {1,2,3,5,4}, {2,3,4,1,5}, {3,4,5,2,1}, {4,5,1,3,2}, {5,1,2,4,3}. Since within each set we allow for rotation ({1,2,3,5,4} and {2,3,5,4,1} are different) but not reflection ({1,2,3,5,4} and {4,5,3,2,1} are counted as one sequence) the total number of ways is 5*5 and therefore T(5,3)=25. Unfolded table with n individuals (rows) forming k pairs (columns):     0    1    2    3    4    5    6    7 0   1 1   0    1 2   0    0    1 3   0    0    0    3 4   0    0    8    0    4 5   5    0   25   25    0    5 6  18   72   90  120   54    0   6 7 161  490  784  637  343   98   0   7 PROG (Java) See Links section. CROSSREFS Cf. A001710 sum of each row. Cf. A326390 (with reflection symmetry), A326404 (with reflection symmetry, but disregards circular symmetry), A326411 (disregards both circular and reflection symmetry). Sequence in context: A243163 A209490 A346879 * A140577 A068606 A106153 Adjacent sequences:  A326394 A326395 A326396 * A326398 A326399 A326400 KEYWORD nonn,tabl AUTHOR Witold Tatkiewicz, Aug 01 2019 STATUS approved

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