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 A255982 Number T(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, such that each axis is used at least once; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 16
 1, 0, 1, 0, 2, 4, 0, 5, 29, 30, 0, 14, 184, 486, 336, 0, 42, 1148, 5880, 9744, 5040, 0, 132, 7228, 64464, 192984, 230400, 95040, 0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160, 0, 1430, 300476, 7043814, 51622600, 165293700, 259518600, 196756560, 57657600 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Rows n = 0..135, flattened FORMULA T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i). EXAMPLE A(3,1) = 5:   [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||]. . A(2,2) = 4:   ._______.  ._______.  ._______.  ._______.   |   |   |  |   |   |  |   |   |  |       |   |___|   |  |   |___|  |___|___|  |_______|   |   |   |  |   |   |  |       |  |   |   |   |___|___|  |___|___|  |_______|  |___|___|. . Triangle T(n,k) begins:   1   0,   1;   0,   2,     4;   0,   5,    29,     30;   0,  14,   184,    486,     336;   0,  42,  1148,   5880,    9744,    5040;   0, 132,  7228,  64464,  192984,  230400,   95040;   0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;   ... MAPLE 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: T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k): seq(seq(T(n, k), k=0..n), n=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}]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424. Main diagonal gives A001761. Row sums give A258425. T(2n,n) give A258426. Cf. A237018, A256061, A258427. Sequence in context: A319275 A269011 A274086 * A256061 A323099 A002652 Adjacent sequences:  A255979 A255980 A255981 * A255983 A255984 A255985 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 13 2015 STATUS approved

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Last modified December 2 08:31 EST 2021. Contains 349437 sequences. (Running on oeis4.)