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 A237018 Number A(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
 1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 39, 14, 0, 1, 5, 32, 132, 212, 42, 0, 1, 6, 50, 314, 1080, 1232, 132, 0, 1, 7, 72, 615, 3440, 9450, 7492, 429, 0, 1, 8, 98, 1065, 8450, 40320, 86544, 47082, 1430, 0, 1, 9, 128, 1694, 17604, 124250, 494736, 819154, 303336, 4862, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The g.f. given below is a generalization of formulas given by Murray R. Bremner and Sara Madariaga in A236339 and A236342. According to them A(n,k) also gives the number of distinct monomials of degree n+1 in the universal algebra with k nonassociative binary products {*1,...,*k} related only by the interchange laws from k-category theory: (a *i b) *j (c *i d) = (a *j c) *i (b *j d) for i,j in {1,...,k} and i coeff(series(RootOf(x*(-1)^k=add((-1)^i* binomial(k, i)*(G*x)^(2^(k-i)), i=0..k), G), x, n+1), x, n): seq(seq(A(n, d-n), n=0..d), d=0..10); # second Maple program: b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1, A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2))) end: A:= proc(n, k) option remember; `if`(n=0, 1, -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k)) end: seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n-1, k], Sum[A[j, k]*b[n-j-1, k, t-1], {j, 0, n-2}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[ Binomial[k, j]*(-1)^j*b[n+1, k, 2^j], {j, 1, k}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000108, A236339(n+1), A236342(n+1), A237019, A237020, A237021, A237022, A237023, A237024, A237025. Rows n=0-2 give: A000012, A001477, A001105. Main diagonal gives A237026. Cf. A255982. Sequence in context: A124540 A124550 A306024 * A290605 A292913 A214776 Adjacent sequences: A237015 A237016 A237017 * A237019 A237020 A237021 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Feb 02 2014 STATUS approved

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Last modified August 8 04:35 EDT 2024. Contains 375018 sequences. (Running on oeis4.)