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A292741 Number A(n,k) of partitions of n with k sorts of part 1; square array A(n,k), n>=0, k>=0, read by antidiagonals. 7
1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 2, 1, 4, 10, 11, 5, 2, 1, 5, 17, 31, 24, 7, 4, 1, 6, 26, 69, 95, 50, 11, 4, 1, 7, 37, 131, 278, 287, 104, 15, 7, 1, 8, 50, 223, 657, 1114, 865, 212, 22, 8, 1, 9, 65, 351, 1340, 3287, 4460, 2599, 431, 30, 12, 1, 10, 82, 521, 2459, 8042, 16439, 17844, 7804, 870, 42, 14 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

FORMULA

G.f. of column k: 1/(1-k*x) * 1/Product_{j>=2} (1-x^j).

A(n,k) = Sum_{j=0..n} A002865(j) * k^(n-j).

EXAMPLE

A(1,3) = 3: 1a, 1b, 1c.

A(2,3) = 10: 2, 1a1a, 1a1b, 1a1c, 1b1a, 1b1b, 1b1c, 1c1a, 1c1b, 1c1c.

A(3,2) = 11: 3, 21a, 21b, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.

Square array A(n,k) begins:

  1,  1,   1,    1,     1,     1,      1,      1, ...

  0,  1,   2,    3,     4,     5,      6,      7, ...

  1,  2,   5,   10,    17,    26,     37,     50, ...

  1,  3,  11,   31,    69,   131,    223,    351, ...

  2,  5,  24,   95,   278,   657,   1340,   2459, ...

  2,  7,  50,  287,  1114,  3287,   8042,  17215, ...

  4, 11, 104,  865,  4460, 16439,  48256, 120509, ...

  4, 15, 212, 2599, 17844, 82199, 289540, 843567, ...

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0 or i<2, k^n,

      add(b(n-i*j, i-1, k), j=0..iquo(n, i)))

    end:

A:= (n, k)-> b(n$2, k):

seq(seq(A(n, d-n), n=0..d), d=0..14);

MATHEMATICA

b[0, _, _] = 1; b[n_, i_, k_] := b[n, i, k] = If[i < 2, k^n, Sum[b[n - i*j, i - 1, k], {j, 0, Quotient[n, i]}]];

A[n_, k_] := b[n, n, k];

Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, May 19 2018, translated from Maple *)

CROSSREFS

Columns k=0-2 give: A002865, A000041, A090764.

Rows n=0-2 give: A000012, A001477, A002522, A071568.

Main diagonal gives A292462.

Cf. A003992, A004248, A009998, A051129, A292508, A292622, A292745.

Sequence in context: A034254 A157103 A135966 * A060351 A290252 A076037

Adjacent sequences:  A292738 A292739 A292740 * A292742 A292743 A292744

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 22 2017

STATUS

approved

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Last modified November 14 22:17 EST 2019. Contains 329134 sequences. (Running on oeis4.)