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A214724
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Expansion of e.g.f.: exp( Sum_{n>=0} x^(n^2+1)/(n^2+1) ).
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1
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1, 1, 2, 4, 10, 50, 220, 1240, 6140, 32860, 602200, 5668400, 62030200, 522328600, 4487190800, 62591332000, 715163146000, 30496564010000, 482341877812000, 8342949421288000, 124613700640580000, 1733826182453140000, 36635355834463000000, 597186420007933040000
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OFFSET
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0,3
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COMMENTS
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Conjecture: p | a(n) for n>=p when p is a prime of the form m^2+1 (A002496).
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LINKS
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 50*x^5/5! + 220*x^6/6! +...
where, by definition,
log(A(x)) = x + x^2/2 + x^5/5 + x^10/10 + x^17/17 + x^26/26 + x^37/37 +...
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MATHEMATICA
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With[{m=30}, CoefficientList[Series[Exp[Sum[x^(n^2+1)/(1+n^2), {n, 0, m+ 2}]], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Jan 07 2024 *)
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PROG
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(PARI) {a(n)=n!*polcoeff(exp(sum(k=0, n, x^(k^2+1)/(k^2+1) + x*O(x^n))), n)}
for(n=0, 21, print1(a(n), ", "))
(Magma)
m:=30;
R<x>:=PowerSeriesRing(Rationals(), m+1);
Coefficients(R!(Laplace( Exp((&+[x^(n^2+1)/(n^2+1): n in [0..m+2]])) ))); // G. C. Greubel, Jan 07 2024
(SageMath)
m=30
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(sum(x^(n^2+1)/(n^2+1) for n in range(m+3))) ).egf_to_ogf().list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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