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A130187
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Numerators of rationals r(n) related to the z-sequence of the Sheffer matrix A060821 for Hermite polynomials.
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2
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1, 3, 5, 105, 189, 3465, 19305, 2027025, 3828825, 130945815, 1249937325, 105411381075, 608142583125, 30494006668125, 412685556908625, 191898783962510625, 372509404162520625, 24627010608522196875
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The denominators are A130188.
The z-sequence for the Sheffer matrix (see the W. Lang link under A006233) A060821(n,m) (coefficients of Hermite polynomials) is z(2*k)=0 and z(2*k+1)= -r(k)/2, k>=0, with r(n):= a(n)/A130188(n).
The recurrence for the entries of the first (m=0) column of the Sheffer triangle A006233(n,m)=:H(n,m) is H(0,0):=1, H(n,0)=n*sum(z(m)*H(n-1,m),m=0..n-1), n>=1.
The e.g.f. for the z-sequence is -2*(exp((x^2)/4)-1)/x.
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LINKS
| W. Lang, Rationals, z-sequence.
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FORMULA
| a(n)=numerator(r(n)), n>=0. r(n):=-2*z(2*n+1) (in lowest terms). The e.g.f. of z(n) is given above.
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EXAMPLE
| r(1)=3/4 leads to z(3)=-3/8.
Rationals r(n):
E.g.f. for z-sequence: -2*(exp((x^2)/4)-1)/x = -1/2*x-1/16*x^3-1/192*x^5-1/3072*x^7-...
z-sequence [0, -1/2, 0, -3/8, 0, -5/8, 0, -105/64, 0, -189/32, 0,...
Recurrence, n=4: H(4,0) = 4*(z(1)*(-12)+z(3)*8)= 4*((-1/2)*(-12)+(-3/8)*8)= 4*3=12.
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CROSSREFS
| Sequence in context: A173487 A103081 A003112 * A054266 A054268 A153137
Adjacent sequences: A130184 A130185 A130186 * A130188 A130189 A130190
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jun 01 2007
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