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%I #14 Oct 09 2013 08:59:32
%S 1,3,2,25,9,49,-464,27,-16175,-3237113,-105387,-1737125143,-770720657,
%T -25881785,-1997012608,-135505859252213,-214182958293,
%U -528114253960241,-19467909708875,-595278405326437,-66462260889140083,-180690496141440384775397,-1610254561193224
%N Numerator of Cotesian number C(n,2).
%D Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
%H Vincenzo Librandi, <a href="/A100645/b100645.txt">Table of n, a(n) for n = 2..100</a>
%e 1/6, 3/8, 2/15, 25/144, 9/280, 49/640, -464/14175, 27/2240, -16175/199584, -3237113/87091200, -105387/875875, -1737125143/22353408000, -770720657/5003856000, -25881785/229605376, ... = A100645/A100646 = A002179/A002176 (the latter not being in lowest terms)
%t cn[n_, 0] := Sum[n^j*StirlingS1[n, j]/(j+1), {j, 1, n+1}]/n!; cn[n_, n_] := cn[n, 0]; cn[n_, k_] := 1/n!*Binomial[n, k]*Sum[n^(j+m)*StirlingS1[k, j]* StirlingS1[n-k, m]/((m+1)*Binomial[j+m+1, m+1]), {m, 1, n}, {j, 1, k+1}]; a[n_] := Numerator[cn[n, 2]]; Table[a[n], {n, 2, 24}] (* _Jean-François Alcover_, Oct 08 2013 *)
%Y Cf. A100646.
%Y See A002176 for further references. A diagonal of A100640/A100641.
%K sign,frac
%O 2,2
%A _N. J. A. Sloane_, Dec 05 2004