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A290603
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Numerators in the expansion of the exponential generating function (1/2)*((1 + 3*x)/x)*(1 - (1 + 3*x)^(-4/3)).
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2
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2, -1, 14, -35, 364, -14560, 79040, -1521520, 304304000, -852051200, 24012352000, -2245154912000, 25560225152000, -949379791360000, 114305326879744000, -1643139073896320000, 75777707878512640000, -33493746882302586880000, 193911166160699187200000, -10684505255454525214720000, 1862156630236360108851200000
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OFFSET
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0,1
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COMMENTS
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The denominators are A038500(n+1), n >= 0.
This gives one half of the numerators of the z-sequence for the generalized unsigned Lah number Sheffer matrix Lah[3,2] = A290598.
For Sheffer a- and z-sequences see a W. Lang link under A006232 with the references for the Riordan case, and also the present link for a proof.
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LINKS
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FORMULA
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a(n) = numerator(r(n)) with the rationals r(n) = [x^n/n!] (1/2)*((1 + 3*x)/x)*(1 - (1 + 3*x)^(-4/3)).
2*a(n)/A038500(n+1) = z(3,2;n) = 4 for n = 0, and ((-1)^n/(n+1)*Product_{j=1..n} (1+3*j) = ((-1)^n/(n+1))*A007559(n+1) for n >= 1.
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EXAMPLE
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The rationals z(3,2;n) = 2*a(n)/A038500(n+1) begin:
{4, -2, 28/3, -70, 728, -29120/3, 158080, -3043040, 608608000/9, -1704102400, 48024704000, -4490309824000/3, ...}
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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