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A008969 Triangle of differences of reciprocals of unity.

%I A008969 #22 Jul 08 2025 01:25:46
%S A008969 1,1,3,1,11,7,1,50,85,15,1,274,1660,575,31,1,1764,48076,46760,3661,63,
%T A008969 1,13068,1942416,6998824,1217776,22631,127,1,109584,104587344,
%U A008969 1744835904,929081776,30480800,137845,255,1,1026576,7245893376,673781602752,1413470290176,117550462624,747497920,833375,511
%N A008969 Triangle of differences of reciprocals of unity.
%D A008969 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
%H A008969 Alois P. Heinz, <a href="/A008969/b008969.txt">Rows n = 1..45, flattened</a>
%e A008969 Triangle T(n,k) begins:
%e A008969   1;
%e A008969   1,      3;
%e A008969   1,     11,         7;
%e A008969   1,     50,        85,         15;
%e A008969   1,    274,      1660,        575,        31;
%e A008969   1,   1764,     48076,      46760,      3661,       63;
%e A008969   1,  13068,   1942416,    6998824,   1217776,    22631,    127;
%e A008969   1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255;
%e A008969   ...
%p A008969 T:= (n,k)-> `if`(k<=n, (n-k+2)!^k *
%p A008969      add((-1)^(j+1)*binomial(n-k+2, j)/ j^k, j=1..n-k+2), 0):
%p A008969 seq(seq(T(n,k), k=0..n), n=0..7); # _Alois P. Heinz_, Sep 05 2008
%t A008969 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_ *)
%Y A008969 Cf. A001236, A001237, A001238, A001240, A001241, A001242.
%Y A008969 Columns include A000254, A000424, A001236, A001237, A001238. Right-hand columns include A000225, A001240, A001241, A001242.
%K A008969 nonn,tabl
%O A008969 1,3
%A A008969 _N. J. A. Sloane_