A293140 -id:A293140 - cp's OEIS frontend

A293139 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} (-1)^jx^(ji)/j!.

1, 1, 0, 1, -1, 0, 1, -1, -2, 0, 1, -1, -1, 0, 0, 1, -1, -1, 0, 0, 0, 1, -1, -1, -1, 0, 120, 0, 1, -1, -1, -1, 0, 0, 0, 0, 1, -1, -1, -1, 1, 20, 180, 5040, 0, 1, -1, -1, -1, 1, 20, 180, 0, 0, 0, 1, -1, -1, -1, 1, 19, 150, 1260, 10080, 0, 0, 1, -1, -1, -1, 1, 19

Offset: 0

Seiichi Manyama, Oct 01 2017

			Square array begins:
   1,   1,  1,  1,  1, ...
   0,  -1, -1, -1, -1, ...
   0,  -2, -1, -1, -1, ...
   0,   0,  0, -1, -1, ...
   0,   0,  0,  0,  1, ...
   0, 120,  0, 20, 20, ...

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

Columns k=0..2 give A000007, A293140, A293141.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293135.

A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 6, 0, 24, 24, 0, 120, 240, 0, 720, 1440, 720, 0, 5040, 15120, 5040, 0, 40320, 120960, 80640, 0, 362880, 1451520, 1088640, 0, 3628800, 14515200, 14515200, 3628800, 0, 39916800, 199584000, 199584000, 39916800, 0, 479001600, 2395008000, 3353011200, 958003200

Offset: 0

Geoffrey Critzer, Feb 23 2022

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2;
  0,     6,      6;
  0,    24,     24;
  0,   120,    240;
  0,   720,   1440,   720;
  0,  5040,  15120,  5040;
  0, 40320, 120960, 80640;
  ...

Columns k=0-1 give: A000007, A000142 (for n>=1).
Cf. A088311 (row sums).
T(A000217(n),n) gives A052295.
Cf. A003056, A293140.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand(x*b(n-i, min(i-1, n-i)))*n!/(n-i)!))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 23 2022

nn = 13; Prepend[Map[Prepend[#, 0] &, Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Product[1 + y x^i, {i, 1, nn}], {x, 0, nn}],{x,y}]], 1]], {1}] // Grid

E.g.f.: Product_{i>=1} (1 + y*x^i).

Sum_{k=0..A003056(n)} (-1)^k * T(n,k) = A293140(n). - Alois P. Heinz, Feb 23 2022

cp's OEIS Frontend

A293139 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} (-1)^jx^(ji)/j!.

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A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

Views

Author

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Maple

Mathematica

Formula

cp's OEIS Frontend

A293139 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} (-1)^j*x^(j*i)/j!.

Views

Author

Keywords

Examples

Links

Crossrefs

A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A293139 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>0} Sum_{j=0..k} (-1)^jx^(ji)/j!.