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 A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals. 26
 1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA G.f. of column k: Product_{i>=1} 1/(1-k*x^i). T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i). EXAMPLE A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b]. Square array A(n,k) begins:   1,  1,   1,    1,     1,      1,      1,      1, ...   0,  1,   2,    3,     4,      5,      6,      7, ...   0,  2,   6,   12,    20,     30,     42,     56, ...   0,  3,  14,   39,    84,    155,    258,    399, ...   0,  5,  34,  129,   356,    805,   1590,   2849, ...   0,  7,  74,  399,  1444,   4055,   9582,  19999, ...   0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...   0, 15, 350, 3783, 23604, 102455, 347010, 983535, ... 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: A:= (n, k)-> b(n\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..12); 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]]]]; A[n_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942. Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636. Main diagonal gives A124577. Cf. A144064, A255970, A256130. Sequence in context: A292870 A306704 A091063 * A198793 A085388 A294498 Adjacent sequences:  A246932 A246933 A246934 * A246936 A246937 A246938 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 08 2014 STATUS approved

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Last modified January 21 21:30 EST 2020. Contains 331128 sequences. (Running on oeis4.)