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

0,4

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].

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)

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 *)

Prog

(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

Crossrefs

Cf. A006252, A352070, A352073.

Sequence in context: A359396 A222583 A222374 * A092584 A145400 A002657

Adjacent sequences: A097394 A097395 A097396 * A097398 A097399 A097400

Keyword

sign

Author

Michael Somos, Aug 13 2004

Status

approved