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A066325
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Coefficients of unitary Hermite polynomials He_n(x).
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5
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1, 0, 1, -1, 0, 1, 0, -3, 0, 1, 3, 0, -6, 0, 1, 0, 15, 0, -10, 0, 1, -15, 0, 45, 0, -15, 0, 1, 0, -105, 0, 105, 0, -21, 0, 1, 105, 0, -420, 0, 210, 0, -28, 0, 1, 0, 945, 0, -1260, 0, 378, 0, -36, 0, 1, -945, 0, 4725, 0, -3150, 0, 630, 0, -45, 0, 1, 0, -10395, 0, 17325, 0
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OFFSET
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0,8
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COMMENTS
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Also number of involutions on n labeled elements with k fixed points times (-1)^(number of 2-cycles).
Also called normalized Hermite polynomials.
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pg 89,94 (2.3.41,54).
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LINKS
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Table of n, a(n) for n=0..70.
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials
E. Elizalde, Cosmology: techniques and observations
D. Foata, Une methode combinatoire pour l'\'{e}tude des fonctions sp\'{e}ciales
Index entries for sequences related to Hermite polynomials
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FORMULA
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T(n, k)=(-2)^((k-n)/2)*n!/(k!*((n-k)/2)!). n-k even. 0 otherwise.
E.g.f. (relative to x): A(x, y)=exp(x*y-x^2/2)
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EXAMPLE
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1; 0,1; -1,0,1; 0,-3,0,1; 3,0,-6,0,1; ...
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PROG
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(Sage)
def A066325_row(n):
T = [0]*(n+1)
if n==1: return [1]
for m in (1..n-1):
a, b, c = 1, 0, 0
for k in range(m, -1, -1):
r = a - (k+1)*c
if k < m : T[k+2] = u;
a, b, c = T[k-1], a, b
u = r
T[1] = u;
return T[1:]
for n in (1..11): A066325_row(n) # Peter Luschny, Nov 01 2012
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CROSSREFS
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Row sums: A001464 (with different signs). Row sums of absolute values: A000085. Cf. A060281.
Absolute values are given in A099174. - M. F. Hasler, Oct 08 2012
Sequence in context: A035653 A126595 A179898 * A099174 A137297 A178117
Adjacent sequences: A066322 A066323 A066324 * A066326 A066327 A066328
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KEYWORD
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sign,tabl
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AUTHOR
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Christian G. Bower, Dec 14 2001
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STATUS
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approved
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