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 A262495 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 6, 12, 7, 1, 0, 1, 10, 31, 33, 11, 1, 0, 1, 14, 73, 130, 77, 16, 1, 0, 1, 21, 165, 464, 438, 157, 22, 1, 0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1, 0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE T(3,1) = 1: 3a. T(3,2) = 2: 2a1b, 1a1b1a. T(3,3) = 1: 1a1b1c. T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c. Triangle T(n,k) begins:   1;   0, 1;   0, 1,  1;   0, 1,  2,   1;   0, 1,  4,   4,    1;   0, 1,  6,  12,    7,     1;   0, 1, 10,  31,   33,    11,    1;   0, 1, 14,  73,  130,    77,   16,    1;   0, 1, 21, 165,  464,   438,  157,   22,   1;   0, 1, 29, 357, 1558,  2216, 1223,  289,  29,  1;   0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1; MAPLE b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))     end: A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n\$2, k-1))): T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2017, translated from Maple *) CROSSREFS Columns k=0-10 give: A000007, A057427, A000065, A262445/6 = A320545, A320546, A320547, A320548, A320549, A320550, A320551, A320552. Main diagonal and lower diagonal give: A000012, A000124 (shifted). Row sums give A262496. T(2n,n) gives A262529. Cf. A256130. Sequence in context: A296207 A253628 A102728 * A323174 A295683 A165519 Adjacent sequences:  A262492 A262493 A262494 * A262496 A262497 A262498 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 24 2015 STATUS approved

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Last modified January 29 07:03 EST 2020. Contains 331337 sequences. (Running on oeis4.)