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%I M1569 N0612 #29 Jan 29 2022 01:14:16
%S 2,6,8,90,288,840,17280,28350,89600,598752,87091200,63063000,
%T 402361344000,5003856000,2066448384,976924698750,3766102179840000,
%U 15209113920000,5377993912811520000,1646485441080480,89903156428800000
%N a(n) = LCM of denominators of Cotesian numbers {C(n,k), 0 <= k <= n}.
%C See A100640 for definition of C(n,k).
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 886.
%D Louis Brand, Differential and Difference Equations, 1966, p. 612.
%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 Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 513.
%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 Vincenzo Librandi, <a href="/A002176/b002176.txt">Table of n, a(n) for n = 1..100</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 886.
%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]
%p Define C(n,k) as in A100640, then: A002176:=proc(n) local t1,k; t1:=1; for k from 0 to n do t1:=lcm(t1,denom(C(n,k))); od: t1; end;
%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_] := LCM @@ Table[ Denominator[cn[n, k]], {k, 0, n}]; Table[a[n], {n, 1, 21}] (* _Jean-François Alcover_, Oct 25 2011 *)
%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) A002176(n)= denominator(cn(n))
%Y Cf. A002177-A002179, A100620, A100621, A100640, A100641, A100642.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms and references from _Michael Somos_
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