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 A002370 a(n) = (2*n-1)^2 * a(n-1) - 3*C(2*n-1,3) * a(n-2) for n>1; a(0) = a(1) = 1. (Formerly M4296 N1796) 3
 1, 1, 6, 120, 5250, 395010, 45197460, 7299452160, 1580682203100, 441926274289500, 154940341854097800, 66565404923242024800, 34389901168124209507800, 21034386936107260971255000, 15032296693671903309613950000, 12411582569784462888618434640000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5. I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6. T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..225 A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5. [Annotated scanned copy] T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2. T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages] See Vol. 3, page 282. FORMULA a(n) = (2*n)! * [x^(2*n)] (1-x^2)^(-1/4)*exp(x^2/4). a(n) = 2^n*GAMMA(n+1/2)*A002801(n)/Pi^(1/2) = GAMMA(n+1/2)*hypergeom([1/4, -n],[],-4)/Pi^(1/2) - Mark van Hoeij, Oct 26 2011 a(n) ~ (2*n)! * exp(1/4) * GAMMA(3/4) / (Pi * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Feb 15 2015 MAPLE a:= proc(n) option remember;       `if`(n<2, 1, (2*n-1)^2 * a(n-1) -3*binomial(2*n-1, 3) *a(n-2))     end: seq(a(n), n=0..20); MATHEMATICA a[n_] := Gamma[n+1/2]*HypergeometricPFQ[{1/4, -n}, {}, -4]/Sqrt[Pi]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 17 2014, after Mark van Hoeij *) PROG (PARI) x='x+O('x^50);  v=Vec( (1-x)^(-1/4)*exp(x/4) ); vector(#v, n, v[n]*(2*n-2)! ) /* show terms */ CROSSREFS Cf. A167028. Sequence in context: A094273 A094278 A093910 * A012846 A012641 A012795 Adjacent sequences:  A002367 A002368 A002369 * A002371 A002372 A002373 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Jon E. Schoenfield, Mar 24 2010 Edited by Alois P. Heinz, Jan 21 2011 STATUS approved

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