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 A256130 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 17
 1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 7, 1, 0, 7, 30, 33, 11, 1, 0, 11, 72, 130, 77, 16, 1, 0, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS In general, column k>1 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - Vaclav Kotesovec, Jun 01 2015 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,k) = A255970(n,k)/k! = (Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i)) / A000142(k). EXAMPLE T(3,1) = 3: 1a1a1a, 2a1a, 3a. T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b. T(3,3) = 1: 1a1b1c. Triangle T(n,k) begins:   1;   0,  1;   0,  2,    1;   0,  3,    4,     1;   0,  5,   12,     7,     1;   0,  7,   30,    33,    11,     1;   0, 11,   72,   130,    77,    16,     1;   0, 15,  160,   463,   438,   157,    22,    1;   0, 22,  351,  1557,  2216,  1223,   289,   29,   1;   0, 30,  743,  5031, 10422,  8331,  2957,  492,  37,  1;   0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46,  1; MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))     end: T:= (n, k)-> add(b(n\$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k): seq(seq(T(n, k), k=0..n), n=0..10); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465. Row sums give A258466. T(2n,n) give A258467. Cf. A000142, A246935, A255970, A262495, A319730. Sequence in context: A285072 A300454 A155112 * A257566 A188286 A101603 Adjacent sequences:  A256127 A256128 A256129 * A256131 A256132 A256133 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 15 2015 STATUS approved

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Last modified December 15 00:30 EST 2019. Contains 329988 sequences. (Running on oeis4.)