A007296 Reversion of (1 + g.f. for primes).

1, -2, 5, -15, 52, -200, 827, -3596, 16191, -74702, 350794, -1669439, 8029728, -38963552, 190499461, -937550897, 4641253152, -23096403422, 115475977145, -579799302750, 2922325238788, -14780595276064, 74995317703482, -381625745964018, 1947147485751919

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• sign
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Transforms
• Index entries for reversions of series

Crossrefs

Cf. A334263.

Programs

Maple

read transforms; s1 := [seq(ithprime(i),i=1..40)]; s2 := [1,op(%)]; REVERT(%);
# Alternative, using function CompInv from A357588.
CompInv(25, n -> if n = 1 then 1 else ithprime(n-1) fi); # Peter Luschny, Oct 05 2022

Mathematica

nmax = 25; Rest[CoefficientList[InverseSeries[Series[x + Sum[Prime[k-1]*x^k, {k, 2, nmax}], {x, 0, nmax}], x], x]] (* Vaclav Kotesovec, Apr 21 2020 *)

Formula

a(n) ~ -(-1)^n / (sqrt(2*Pi*t) * n^(3/2) * r^(n - 1/2)), where t = Sum_{k>=0} (k+1)*(k+2)*prime(k+1) * s^k = 2.76855665284448835155556293964568965050630014..., s = -0.4018472849329562729164121279063799981049446018535... is the root of the equation Sum_{k>=1} (k+1)*prime(k) * s^k = -1 and r = -s - Sum_{k>=2} prime(k-1) * s^k = 0.18422249999982341975449666640383532448650252568... - Vaclav Kotesovec, Apr 21 2020

Extensions

Signs corrected Dec 24 2001