|
%I M3165 N1283 #43 Oct 18 2022 19:14:32
%S 3,60,630,5040,34650,216216,1261260,7001280,37413090,193993800,
%T 981608628,4867480800,23728968900,114011377200,540972351000,
%U 2538963567360,11802213457650,54396360988200,248812984520100,1130341536324000,5103492036502860,22913637714910800
%N Coefficients for extrapolation.
%C Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-2 of B equals a(n-3). - _T. D. Noe_, May 01 2011
%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
%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 G. C. Greubel, <a href="/A002738/b002738.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%F From _Alois P. Heinz_, May 02 2011: (Start)
%F a(n) = 3*binomial(2*n+3,n)*binomial(n+3,n).
%F G.f.: 3*(1 + 6*x)/(1-4*x)^(7/2). (End)
%F a(n) = binomial(2*n+3,n)*(n^3 + 6*n^2 + 11*n+6)/2. - _Charles R Greathouse IV_, May 02 2011
%F a(n) = 3*A007744(n). - _R. J. Mathar_, Jan 21 2020
%F a(n) = (3/2)*( 5*A020918(n) - 3*A002802(n)). - _G. C. Greubel_, Mar 21 2022
%t Table[Total[Inverse[HilbertMatrix[n]][[n - 2]]], {n, 3, 25}] (* _T. D. Noe_, May 02 2011 *)
%o (Magma) [3*Binomial(2*n+3,n)*Binomial(n+3,3): n in [0..30]]; // _G. C. Greubel_, Mar 21 2022
%o (Sage) [3*binomial(2*n+3,3)*binomial(2*n,n) for n in (0..30)] # _G. C. Greubel_, Mar 21 2022
%Y Cf. A002802, A007744, A020918.
%Y A diagonal of A331431.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Extended by _T. D. Noe_, May 01 2011
| 