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 A183157 Triangle read by rows: T(n,k) is the number of partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|). 2
 1, 1, 1, 1, 4, 2, 1, 9, 10, 2, 1, 16, 28, 12, 2, 1, 25, 60, 40, 14, 2, 1, 36, 110, 100, 54, 16, 2, 1, 49, 182, 210, 154, 70, 18, 2, 1, 64, 280, 392, 364, 224, 88, 20, 2, 1, 81, 408, 672, 756, 588, 312, 108, 22, 2, 1, 100, 570, 1080, 1428, 1344, 900, 420, 130, 24, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Rows also give the coefficients of the clique polynomial of the n X n bishop graph. - Eric W. Weisstein, Jun 04 2017 LINKS R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011. Eric Weisstein's World of Mathematics, Bishop Graph Eric Weisstein's World of Mathematics, Clique Polynomial FORMULA T(n,0)=1, T(n,1) = n^2 and T(n,k)=2*(2*n-k+1)*binomial(n,k)/(k+1), k > 1. EXAMPLE T (3,2) = 10 because there are exactly 10 partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,1); (1,2)-->(2,3); (1,2)-->(3,2); (2,3)-->(1,2); (2,3)-->(2,1); (2,3)-->(2,3); (2,3)-->(3,2); (1,3)-->(1,3); (1,3)-->(3,1) - the mappings are coordinate-wise. The triangle starts   1;   1,    1;   1,    4,    2;   1,    9,   10,    2;   1,   16,   28,   12,    2;   1,   25,   60,   40,   14,    2;   1,   36,  110,  100,   54,   16,    2;   1,   49,  182,  210,  154,   70,   18,    2;   1,   64,  280,  392,  364,  224,   88,   20,    2;   1,   81,  408,  672,  756,  588,  312,  108,   22,    2;   1,  100,  570, 1080, 1428, 1344,  900,  420,  130,   24,    2; MAPLE A183157 := proc(n, k) if k =0 then 1; elif k = 1 then n^2 ; else 2*(2*n-k+1)*binomial(n, k)/(k+1) ; end if; end proc: # R. J. Mathar, Jan 06 2011 MATHEMATICA T[_, 0] = 1; T[n_, 1] := n^2; T[n_, k_] := 2*(2*n - k + 1)*Binomial[n, k] / (k + 1); Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 25 2017 *) CROSSREFS Cf. A183156 (row sums), A006331 (k=2), A008911 (k=3), A067056 (k=4). Sequence in context: A101020 A160905 A208612 * A211957 A063983 A259985 Adjacent sequences:  A183154 A183155 A183156 * A183158 A183159 A183160 KEYWORD nonn,tabl AUTHOR Abdullahi Umar, Dec 28 2010 STATUS approved

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)