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A160068
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Numerator of Hermite(n, 24/25).
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1
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1, 48, 1054, -69408, -7284084, -2596032, 45400915464, 2198714182272, -291719729560944, -35989688841645312, 1554341893161645024, 524479521392325361152, 3802815995858998255296, -7684657653083648501025792, -430659327280723849697798016
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OFFSET
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0,2
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LINKS
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FORMULA
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Theorem: E.g.f.: exp(-x*(625*x-48)); a(n+2) = -1250*(n+1)*a(n)+48*a(n+1).
Proof:
The Hermite polynomials H(n,t) have e.g.f. g(x) = exp(-x^2 + 2 x t).
Thus b(n) := H(n, 24/25)*5^(2n) has e.g.f. exp(-x*(625*x -48)) and satisfies the recurrence b(n+2) = -1250*(n+1)*b(n)+48*b(n+1), n>=0.
To show that these are the numerators of H(n,24/25), we need to show that b(n) is never divisible by 5. But taking the recurrence mod 5 we get b(n+2) == 3*b(n+1) mod 5.
Since b(0) and b(1) are not divisible by 5, induction finishes the proof.
(End)
a(n) = 25^n * Hermite(n, 24/25).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(48/25)^(n-2*k)/(k!*(n-2*k)!)). (End)
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EXAMPLE
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Numerators of 1, 48/25, 1054/625, -69408/15625, -7284084/390625, ...
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MAPLE
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MATHEMATICA
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Table[25^n*HermiteH[n, 24/25], {n, 0, 30}] (* G. C. Greubel, Jul 11 2018 *)
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PROG
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(Magma) [Numerator((&+[(-1)^k*Factorial(n)*(48/25)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 11 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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