A097397 Coefficients in asymptotic expansion of normal probability function.

1, 1, 1, 5, 9, 129, 57, 9141, -36879, 1430049, -15439407, 418019205, -7404957255, 196896257505, -4656470025015, 134136890777205, -3845524501226655, 123250625100419265, -4085349586734306015, 145973136800663973765

Offset: 0

Author

Michael Somos, Aug 13 2004

Keywords

• sign

Comments

a(0) + a(1)*x/(1-2*x) + a(2)*x^2/((1-2*x)*(1-4*x)) + ... = 1 + x + 3*x^2 + 15*x^3 + ...

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 932.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A006252, A352070, A352073.

Programs

Mathematica

Table[Sum[2^(n - 2*k)*(2*k)!/k! * SeriesCoefficient[(1 - n + x)*Pochhammer[2 - n + x, -1 + n], {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 10 2019 *)

PARI

a(n)=sum(k=0,n, 2^(n-2*k)*(2*k)!/k!* polcoeff(prod(i=0,n-1,x-i),k))

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-log(1+2*x)))) \\ Seiichi Manyama, Mar 05 2022

Formula

E.g.f.: 1/sqrt(1 - log(1 + 2*x)). - Seiichi Manyama, Mar 05 2022

a(n) ~ n! * (-1)^(n+1) * 2^(n-1) / (log(n)^(3/2) * n) * (1 - 3*(gamma + 1)/(2*log(n)) + 15*(1 + 2*gamma + gamma^2 - Pi^2/6) / (8*log(n)^2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 05 2022

From Seiichi Manyama, Nov 18 2023: (Start)

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (1/2 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). (End)