A008969 Triangle read by rows of differences of reciprocals of unity.

1, 1, 3, 1, 11, 7, 1, 50, 85, 15, 1, 274, 1660, 575, 31, 1, 1764, 48076, 46760, 3661, 63, 1, 13068, 1942416, 6998824, 1217776, 22631, 127, 1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255, 1, 1026576, 7245893376, 673781602752, 1413470290176, 117550462624, 747497920, 833375, 511

Offset

1,3

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.

Links

Alois P. Heinz, Rows n = 1..45, flattened

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, page 228 annotated with A-numbers by N. J. A. Slonae circa 1967, and scanned.

Formula

Conjecture: T(n, k) is the determinant of the (n-k+1) X (n-k+1) matrix, where the (i, j)-th entry is Stirling2(k + i + 1, j + 1) for 0 <= k <= n. - Mikhail Kurkov, Jun 01 2026

Example

Triangle T(n,k) begins:
  1;
  1,      3;
  1,     11,         7;
  1,     50,        85,         15;
  1,    274,      1660,        575,        31;
  1,   1764,     48076,      46760,      3661,       63;
  1,  13068,   1942416,    6998824,   1217776,    22631,    127;
  1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255;
  ...

Maple

T:= (n, k)-> `if`(k<=n, (n-k+2)!^(k-1) *
     add((-1)^(j+1)*binomial(n-k+2, j)/ j^(k-1), j=1..n-k+2), 0):
seq(seq(T(n, k), k=1..n), n=1..9);  # Alois P. Heinz, Sep 05 2008
# Alternative:
S := proc(n, r, j) option remember;
        if n = 0 then 1/j^r;
        else S(n-1, r, j+1)-S(n-1, r, j)
        end if;
end proc:
T := (n, r) -> ((n+1)!)^r * abs(S(n, r, 1)) / n!:
seq( seq(T(n+1-k, k), k=1..n) , n=1..9);
# Example: > T(10, 12):
# 4498607227742743734699578157092398416324947564107251882145271134895996928000000000000 # Brendan McKay, Oct 15 2025

Mathematica

T[n_, k_] := If[k <= n, (n-k+2)!^k*Sum[(-1)^(j+1)*Binomial[n-k+2, j]/j^k, {j, 1, n-k+2}], 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

Crossrefs

Cf. A001236, A001237, A001238, A001240, A001241, A001242.

Columns include A000254, A000424, A001236, A001237, A001238.

Right-hand columns include A000225, A001240, A001241, A001242.

Sequence in context: A111965 A110440 A135574 * A199577 A228534 A119908

Adjacent sequences: A008966 A008967 A008968 * A008970 A008971 A008972

Keyword

nonn,tabl,changed

Author

N. J. A. Sloane

Status

approved