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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-Fran├žois 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: A130020 A292870 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 October 20 11:17 EDT 2018. Contains 316379 sequences. (Running on oeis4.)