A275429 -id:A275429 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Jul 27 2016

			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, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition of a set

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

Main diagonal gives A275429.

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);

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-François Alcover, Feb 08 2017, translated from Maple *)

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

A335797 a(n) = n! * [x^n] exp(Sum_{k=1..n, gcd(n,k) = 1} x^k / k!).

Original entry on oeis.org

1, 1, 1, 4, 5, 51, 7, 876, 457, 7678, 5271, 678569, 10705, 27644436, 5060161, 133924576, 197920145, 82864869803, 173283535, 5832742205056, 98269310261, 34660429169122, 25313714237505, 44152005855084345, 13685698802401, 2410161938206898126, 129066382491033573

Offset: 0

Ilya Gutkovskiy, Oct 12 2020

nonn

Number of set partitions of [n] into blocks that are relatively prime to n.

Alois P. Heinz, Table of n, a(n) for n = 0..576

Cf. A057562, A275429, A335088.

b:= proc(n, m) option remember; `if`(n=0, 1, add(`if`(
      igcd(j, m)=1, b(n-j, m), 0)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..27);  # Alois P. Heinz, Oct 12 2020

Table[n! SeriesCoefficient[Exp[Sum[Boole[GCD[n, k] == 1] x^k/k!, {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

A332259 a(n) = n! * [x^n] 1 / (1 - Sum_{d|n} x^d / d!).

Original entry on oeis.org

1, 1, 3, 7, 67, 121, 4551, 5041, 405371, 888721, 65326213, 39916801, 27854708575, 6227020801, 5417436748153, 6968620334677, 2744261072866171, 355687428096001, 2984245819328278077, 121645100408832001, 1177257398964663961517, 545405274481512519361

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Number of ordered set partitions of [n] such that n is a multiple of each block size.

Cf. A000670, A275429.

Table[n! SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] == 0] x^k/k!, {k, 1, n}]), {x, 0, n}], {n, 0, 22}]

a(n) = {n! * polcoef(1/(1 - sumdiv(n, d, x^d/d!) + O(x*x^n)), n)} \\ Andrew Howroyd, Feb 08 2020

cp's OEIS Frontend

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.

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A335797 a(n) = n! * [x^n] exp(Sum_{k=1..n, gcd(n,k) = 1} x^k / k!).

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A332259 a(n) = n! * [x^n] 1 / (1 - Sum_{d|n} x^d / d!).

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