A394559 Triangle read by rows: inverse transform relative to set partitions into 3-tori (see A393519).

1, 0, 1, 0, 3, 1, 0, 21, 9, 1, 0, 261, 111, 18, 1, 0, 4743, 1935, 345, 30, 1, 0, 112779, 44613, 8100, 825, 45, 1, 0, 3297213, 1280097, 232533, 25200, 1680, 63, 1, 0, 114135093, 43813143, 7946694, 885213, 64890, 3066, 84, 1, 0, 4561146639, 1738235295, 315065565, 35624610, 2745603, 146286, 5166, 108, 1

Offset

0,5

Comments

If B2(n) = Sum_{k=0..n} V(n, k) * (-1)^(n-k) * B(k) where V(n, k) is A393519 then B(n) = Sum_{k=0..n} T(n, k) * B2(k) where T(n, k) is this sequence (pair of sequence transforms).

Links

Table of n, a(n) for n=0..54.

Marko Riedel, Maple code for sequence.

Marko Riedel, Definitions and auxiliary material.

Formula

E.g.f. T(n, k) = n! [w^n] G(w)^k/k! where G(w) is given at A394560, i.e G(w) = Sum_{m>= 0} A394560(m)*w^m/m!.

The subdiagonal is A045943 (consider the linear term and the quadratic term of G(w)).

T(n, k) = (1/k) * Sum_{j=1..n-k+1} binomial(n, j) * A394560(j) * T(n - j, k - 1) for k > 0, T(n, 0) = 0^n. - Peter Luschny, Mar 27 2026

Example

  n\k| 0     1     2    3   4  5
  ---+--------------------------
   0 | 1;
   1 | 0,    1;
   2 | 0,    3,    1;
   3 | 0,   21,    9,   1;
   4 | 0,  261,  111,  18,  1;
   5 | 0, 4743, 1935, 345, 30, 1;

Prog

(SageMath)
def A394559Triangle(N: int) -> list[list[int]]:
    seq = LagrangeInversion(A007425, 19)  # defined in A394560
    return BellTransform(seq) # defined in A394439.
for row in A394559Triangle(9): print(row)  # Peter Luschny, Mar 27 2026

Crossrefs

Cf. A394560 (column 1), A393519, A045943 (subdiagonal), A394595 (row sums), A394439.

Sequence in context: A246049 A316773 A394474 * A006837 A158782 A187558

Adjacent sequences: A394556 A394557 A394558 * A394560 A394561 A394562

Keyword

nonn,tabl

Author

Marko Riedel, Mar 24 2026

Status

approved