A361380 -id:A361380 - cp's OEIS frontend

A347420 Number of partitions of [n] where the first k elements are marked (0 <= k <= n) and at least k blocks contain their own index.

1, 2, 5, 14, 45, 164, 667, 2986, 14551, 76498, 430747, 2582448, 16403029, 109918746, 774289169, 5715471606, 44087879137, 354521950932, 2965359744447, 25749723493074, 231719153184019, 2157494726318234, 20753996174222511, 205985762120971168, 2106795754056142537

Offset: 0

Alois P. Heinz, Jan 05 2022

nonn

			a(3) = 14 = 5 + 5 + 3 + 1: 123, 12|3, 13|2, 1|23, 1|2|3, 1'23, 1'2|3, 1'3|2, 1'|23, 1'|2|3, 1'3|2', 1'|2'3, 1'|2'|3, 1'|2'|3'.

Alois P. Heinz, Table of n, a(n) for n = 0..575
Wikipedia, Partition of a set

Antidiagonal sums of A108087.

Cf. A000110, A005490, A059841, A259691, A350589, A361380.

b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
a:= n-> add(b(i, n-i), i=0..n):
seq(a(n), n=0..25);

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]];
a[n_] := Sum[b[i, n - i], {i, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} A108087(n-k,k).
a(n) = 1 + A005490(n).
a(n) = A000110(n) + Sum_{k=1..n} k * A259691(n-1,k).
a(n) = Sum_{k=1..n} (k+1) * A259691(n-1,k).
a(n) = A000110(n) + A350589(n).
a(n) mod 2 = A059841(n).

A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 5, 1, -2, 2, 1, 15, 1, -3, 5, -3, 4, 52, 1, -4, 10, -13, 7, 11, 203, 1, -5, 17, -35, 36, -10, 41, 877, 1, -6, 26, -75, 127, -101, 31, 162, 4140, 1, -7, 37, -139, 340, -472, 293, -21, 715, 21147, 1, -8, 50, -233, 759, -1573, 1787, -848, 204, 3425, 115975

Offset: 0

Alois P. Heinz, Mar 23 2023

			Square array A(n,k) begins:
    1,   1,   1,    1,     1,      1,       1,       1, ...
    1,   0,  -1,   -2,    -3,     -4,      -5,      -6, ...
    2,   1,   2,    5,    10,     17,      26,      37, ...
    5,   1,  -3,  -13,   -35,    -75,    -139,    -233, ...
   15,   4,   7,   36,   127,    340,     759,    1492, ...
   52,  11, -10, -101,  -472,  -1573,   -4214,   -9685, ...
  203,  41,  31,  293,  1787,   7393,   23711,   63581, ...
  877, 162, -21, -848, -6855, -35178, -134873, -421356, ...

Alois P. Heinz, Antidiagonals n = 0..150, flattened

Columns k=0-5 give: A000110, A000296, A126617, A346738, A346739, A346740.
Rows n=0-2 give: A000012, A024000, A160457.
Main diagonal gives A290219.
Antidiagonal sums give A361380.
Cf. A108087.

T:= func< n,k | (&+[(-k)^j*Binomial(n,j)*Bell(n-j): j in [0..n]]) >;
A361781:= func< n,k | T(k, n-k) >;
[A361781(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024

A:= proc(n, k) option remember; uses combinat;
      add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
A:= (n, k)-> b(n, -k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

T[n_, k_]:= T[n, k]= If[k==0, BellB[n], Sum[(-k)^j*Binomial[n,j]*BellB[n-j], {j,0,n}]];
A361781[n_, k_]= T[k, n-k];
Table[A361781[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 12 2024 *)

def T(n,k): return sum( (-k)^j*binomial(n,j)*bell_number(n-j) for j in range(n+1))
def A361781(n, k): return T(k, n-k)
flatten([[A361781(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

E.g.f. of column k: exp(exp(x) - k*x - 1).

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

cp's OEIS Frontend

A347420 Number of partitions of [n] where the first k elements are marked (0 <= k <= n) and at least k blocks contain their own index.

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