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A159858
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Numerator of Hermite(n, 1/23).
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5
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1, 2, -1054, -6340, 3332716, 33496312, -17563075016, -247760738608, 129576612091280, 2356200115760672, -1229116100101646816, -27386829424951203392, 14249679514133063237824, 376202545407446604740480, -195237686946571258563550336, -5962787476468241626543090432
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OFFSET
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0,2
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COMMENTS
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Consider any odd integer k. B(n) = k^n Hermite(n,1/k) satisfies the recurrence B(n) = 2*B(n-1) - 2*k^2*(n-1)*B(n-2) with B(0) = 1 and B(1) = 2. In particular, B(n) == 2*B(n-1) mod k, and B(n) is coprime to k. Therefore B(n) is the numerator of Hermite(n,1/k). - Robert Israel, Jun 27 2014
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LINKS
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FORMULA
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D-finite with recurrence a(n) - 2*a(n-1) + 1058*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014
a(n) = 23^n * Hermite(n,1/23). This does satisfy the above formula. - Robert Israel, Jun 27 2014
E.g.f.: exp(2*x-529*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(2/23)^(n-2*k)/(k!*(n-2*k)!)). (End)
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EXAMPLE
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Numerator of 1, 2/23, -1054/529, -6340/12167, 3332716/279841, 33496312/6436343, -17563075016/148035889, -247760738608/3404825447, 129576612091280/78310985281...
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MAPLE
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orthopoly[H](n, 1/23) ;
numer(%) ;
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MATHEMATICA
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PROG
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(Magma) [Numerator((&+[(-1)^k*Factorial(n)*(2/23)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018
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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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STATUS
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approved
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