A113775 -id:A113775 - cp's OEIS frontend

A293525 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. Product_{j > 0, j mod k > 0} exp(x^j).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 7, 0, 1, 1, 3, 7, 25, 0, 1, 1, 3, 13, 49, 181, 0, 1, 1, 3, 13, 49, 321, 1201, 0, 1, 1, 3, 13, 73, 381, 2131, 10291, 0, 1, 1, 3, 13, 73, 381, 2971, 19783, 97777, 0, 1, 1, 3, 13, 73, 501, 3331, 26713, 195777, 1013545, 0, 1, 1

Offset: 0

Seiichi Manyama, Oct 11 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   1,   3,   3,   3, ...
   0,   7,   7,  13,  13, ...
   0,  25,  49,  49,  73, ...
   0, 181, 321, 381, 381, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..3 give A000007, A088009, A113775.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293530.

kmax = 12; col[k_] := PadRight[(Exp[Sum[x^j, {j, 1, k - 1}]/(1 - x^k)] + O[x]^kmax // CoefficientList[#, x] &), kmax]*Range[0, kmax - 1]!; A = Array[col, kmax]; Table[A[[n - k + 1, k]], {n, 1, kmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 12 2017, from formula *)

E.g.f. of column k: exp((Sum_{j=1..k-1} x^j)/(1 - x^k)).

A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

Original entry on oeis.org

1, 1, 1, 7, 25, 141, 1171, 9913, 85233, 907273, 11010691, 143824341, 1988010553, 29605763773, 475664908083, 8284952367721, 153508912353121, 2997209814190353, 61485486404453443, 1326994255131585373, 30144049509450774441, 718905298680190094341, 17940822818538396541843

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 7 counts: {(1),(2),(3)}, {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.

b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
      add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 23 2025

With[{m = 22}, CoefficientList[1 + Series[Sum[((x - x^(i + 1))/(1 - x))^i/i!, {i, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

R_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(sum(i=0,N,((x-x^(i+1))/(1-x))^i/i!)))}

E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.

A386497 Number of sets of lists of [n] such that one list is the largest.

Original entry on oeis.org

1, 1, 2, 12, 60, 440, 3390, 33852, 338072, 4116240, 51776730, 736751180, 11075784852, 183142075272, 3157190863190, 59336602681020, 1164223828582320, 24348331444705952, 533422896546272562, 12365952739192923660, 298208300418298756460, 7570420981014167756760

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A088009, A088311, A088312, A088313, A097979, A113775, A351884, A386474.

b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),
      `if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 23 2025

With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1,N, x^j*exp((x-x^j)/(1-x)))))}

E.g.f.: 1 + Sum_{j>0} x^j * exp((x - x^j)/(1 - x)).

cp's OEIS Frontend

A293525 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. Product_{j > 0, j mod k > 0} exp(x^j).

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A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

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Maple

Mathematica

PARI

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A386497 Number of sets of lists of [n] such that one list is the largest.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula