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 A296548 Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n. 1
 1, 1, 1, 1, 6, 1, 1, 28, 28, 1, 1, 120, 560, 120, 1, 1, 496, 9920, 9920, 496, 1, 1, 2016, 166656, 714240, 166656, 2016, 1, 1, 8128, 2731008, 48377856, 48377856, 2731008, 8128, 1, 1, 32640, 44216320, 3183575040, 13158776832, 3183575040, 44216320, 32640, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Equivalently, T(n,k) is the number of n X n matrices, P, over GF(2) with rank k, such that P^2 = P. Equivalently, T(n,k) is the number of direct sum decompositions of the vector space GF(2)^n into exactly two subspaces U and W such that the dimension of U is k. LINKS Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018. Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1. FORMULA T(n,k)/A002884(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A002884(n) * Sum_{n>=0} y*x^n\A002884(n). T(n,k) = A002884(n)/(A002884(k)*A002884(n-k)) = A022166(n,k)*2^(k(n-k)). EXAMPLE Triangle begins:   1;   1,    1;   1,    6,      1;   1,   28,     28,      1;   1,  120,    560,    120,      1;   1,  496,   9920,   9920,    496,    1;   1, 2016, 166656, 714240, 166656, 2016, 1; MATHEMATICA nn = 8; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[     QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,   Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}] Sum[z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]] CROSSREFS Cf. A132186 (row sums). Sequence in context: A166960 A155908 A105373 * A201461 A265603 A174186 Adjacent sequences:  A296545 A296546 A296547 * A296549 A296550 A296551 KEYWORD nonn,tabl AUTHOR Geoffrey Critzer, Dec 15 2017 STATUS approved

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Last modified July 19 04:25 EDT 2019. Contains 325144 sequences. (Running on oeis4.)