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A275422 Number A(n,k) of set partitions of [n] such that k is a multiple of each block size; square array A(n,k), n>=0, k>=0, read by antidiagonals. 10
1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 15, 1, 1, 1, 4, 1, 52, 1, 1, 2, 2, 10, 1, 203, 1, 1, 1, 4, 5, 26, 1, 877, 1, 1, 2, 1, 11, 11, 76, 1, 4140, 1, 1, 1, 5, 1, 31, 31, 232, 1, 21147, 1, 1, 2, 1, 14, 2, 106, 106, 764, 1, 115975, 1, 1, 1, 4, 1, 46, 7, 372, 337, 2620, 1, 678570 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Wikipedia, Partition of a set

FORMULA

E.g.f. for column k>0: exp(Sum_{d|k} x^d/d!), for k=0: exp(exp(x)-1).

EXAMPLE

A(5,3) = 11: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345, 1|2|3|4|5.

A(4,4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

A(6,5) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.

Square array A(n,k) begins:

:    1, 1,   1,   1,    1,  1,    1, 1,    1, ...

:    1, 1,   1,   1,    1,  1,    1, 1,    1, ...

:    2, 1,   2,   1,    2,  1,    2, 1,    2, ...

:    5, 1,   4,   2,    4,  1,    5, 1,    4, ...

:   15, 1,  10,   5,   11,  1,   14, 1,   11, ...

:   52, 1,  26,  11,   31,  2,   46, 1,   31, ...

:  203, 1,  76,  31,  106,  7,  167, 1,  106, ...

:  877, 1, 232, 106,  372, 22,  659, 2,  372, ...

: 4140, 1, 764, 337, 1499, 57, 2836, 9, 1500, ...

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1, add(

      `if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=

      `if`(k=0, 1..n, numtheory[divisors](k))))

    end:

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

MATHEMATICA

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[j>n, 0, A[n-j, k]*Binomial[n-1, j - 1]], {j, If[k==0, Range[n], Divisors[k]]}]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Feb 08 2017, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000110, A000012, A000085, A190865, A190452, A275423, A275424, A275425, A275426, A275427, A275428.

Main diagonal gives A275429.

Sequence in context: A213945 A290771 A014651 * A169951 A174453 A082063

Adjacent sequences:  A275419 A275420 A275421 * A275423 A275424 A275425

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 27 2016

STATUS

approved

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Last modified February 23 06:13 EST 2020. Contains 332159 sequences. (Running on oeis4.)