A394559 - OEIS

%I #46 Apr 23 2026 04:15:34

%S 1,0,1,0,3,1,0,21,9,1,0,261,111,18,1,0,4743,1935,345,30,1,0,112779,

%T 44613,8100,825,45,1,0,3297213,1280097,232533,25200,1680,63,1,0,

%U 114135093,43813143,7946694,885213,64890,3066,84,1,0,4561146639,1738235295,315065565,35624610,2745603,146286,5166,108,1

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

%C 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).

%H Marko Riedel, <a href="/A394559/a394559.maple.txt">Maple code for sequence</a>.

%H Marko Riedel, <a href="/A394559/a394559_1.pdf">Definitions and auxiliary material</a>.

%F 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!.

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

%F 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

%e   n\k| 0     1     2    3   4  5

%e   ---+--------------------------

%e    0 | 1;

%e    1 | 0,    1;

%e    2 | 0,    3,    1;

%e    3 | 0,   21,    9,   1;

%e    4 | 0,  261,  111,  18,  1;

%e    5 | 0, 4743, 1935, 345, 30, 1;

%o (SageMath)

%o def A394559Triangle(N: int) -> list[list[int]]:

%o     seq = LagrangeInversion(A007425, 19)  # defined in A394560

%o     return BellTransform(seq) # defined in A394439.

%o for row in A394559Triangle(9): print(row)  # _Peter Luschny_, Mar 27 2026

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

%K nonn,tabl

%O 0,5

%A _Marko Riedel_, Mar 24 2026