A002177 - OEIS

%I M4364 N1829 #26 Jan 29 2022 01:14:29

%S 1,1,1,7,19,41,751,989,2857,16067,2171465,1364651,8181904909,90241897,

%T 35310023,15043611773,55294720874657,203732352169,69028763155644023,

%U 19470140241329,1022779523247467,396760150748100749

%N Numerators of Cotesian numbers (not in lowest terms): A002176(n)*C(n,0).

%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 Vincenzo Librandi, <a href="/A002177/b002177.txt">Table of n, a(n) for n = 1..100</a>

%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}]; a[n_] := cn[n, 0]*LCM @@ Table[ Denominator[cn[n, k]], {k, 0, n}]; Table[a[n], {n, 1, 22}] (* _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) ncn(n) = denominator(cn(n)) * cn(n);

%o (PARI) nk(n,k) = if(k<0 || k>n, 0, ncn(n)[ k+1 ]);

%o (PARI) A002177(n) = nk(n,0);

%Y Cf. A100640/A100641, A100620/A100621, A002176-A002179.

%K nonn,easy,nice

%O 1,4

%A _N. J. A. Sloane_

%E More terms from _Michael Somos_