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A130408
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Numerators of a-sequence for Sheffer matrix A130191 (Stirling2 squared).
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2
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1, 1, -1, 3, -44, 49, -9895, 3124, -54429, 2624879, -59124785, 163841201, -2508904105349, 1776678914237, -2029995134495, 175211074573961, -21557683580436716, 94127808754677868, -87882971047931164843, 161354083950193175137, -104683178840085862057001
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OFFSET
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0,4
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COMMENTS
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The denominators are found in A130409.
From the definition of the a-sequence {r(n)} one has the recurrence for (Stirling2)^2 = S2sq: S2sq(n,m) = (n/m)*Sum_{j=0..n-m} binomial(m-1+j,j)*r(j)*S2sq(n-1,m-1+j), n >= m >= 1.
For the notion of the a-sequence for a Sheffer matrix see the W. Lang link under A006232. Here the a-sequence is called r(n) because it is a sequence of rationals.
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LINKS
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FORMULA
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a(n) = numerator(r(n)), n >= 0, with the rational r(n) sequence with e.g.f. x/log(1+log(1+x)). {r(n)} is the a-sequence for the Sheffer matrix (Stirling2)^2 (A130191).
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EXAMPLE
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Rationals r(n): [1, 1, -1/3, 3/4, -44/15, 49/3, -9895/84, 3124/3, -54429/5, ...].
Recurrence for (Stirling2)^2: 32=S2sq(4,2) = (4/2)*(1*1*5 + 2*1*6 + 3*(-1/3)*1).
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MAPLE
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seq( numer( coeff(series( x/log(1+log(1+x)), x, n+2)*factorial(n), x, n) ), n = 0..20); # G. C. Greubel, Jan 26 2020
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MATHEMATICA
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With[{m = 20}, CoefficientList[Series[x/Log[1+Log[1+x]], {x, 0, m}], x]*Range[0, m]!]//Numerator (* G. C. Greubel, Jan 26 2020 *)
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PROG
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(Magma) m:=22; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x/Log(1+Log(1+x)) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Jan 26 2020
(Sage) [numerator( factorial(n)*( x/log(1+log(1+x)) ).series(x, n+1).list()[n]) for n in (0..20)] # G. C. Greubel, Jan 26 2020
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CROSSREFS
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KEYWORD
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sign,frac,easy
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AUTHOR
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STATUS
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approved
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