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%I M2921 N1172 #23 Oct 14 2023 21:15:32
%S 0,1,3,12,50,27,1323,-928,1080,-48525,-3237113,-7587864,-31268252574,
%T -770720657,-232936065,-179731134720,-542023437008852,-3212744374395,
%U -926840515700222955,-389358194177500,-17858352159793110
%N Numerators of Cotesian numbers (not in lowest terms): A002176*C(n,2).
%D W. W. Johnson, On Cotesian numbers: their history, computation and values to n=20, Quart. J. Pure Appl. Math., 46 (1914), 52-65.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H W. M. Johnson, <a href="/A002176/a002176.pdf">On Cotesian numbers: their history, computation and values to n=20</a>, Quart. J. Pure Appl. Math., 46 (1914), 52-65. [Annotated scanned copy]
%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}]; A002176[n_] := LCM @@ Table[Denominator[cn[n, k]], {k, 0, n}]; a[2] = 0; a[n_] := A002176[n-1]*cn[n-1, 2]; Table[a[n], {n, 2, 22}] (* _Jean-François Alcover_, Oct 08 2013 *)
%o (PARI) cn(n)= mattranspose(matinverseimage( matrix(n+1,n+1,k,m,(m-1)^(k-1)),matrix(n+1,1,k,m,n^(k-1)/k)))[ 1, ] \\ vector of quadrature formula coefficients via matrix solution
%o (PARI) ncn(n)= denominator(cn(n))*cn(n); nk(n,k)= if(k<0 || k>n,0,ncn(n)[ k+1 ]); A002177(n)= nk(n,2)
%Y Cf. A100640/A100641, A100620/A100621, A002176, A002177, A002178.
%K sign,easy
%O 2,3
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_
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