A154288 Numerators of coefficients in expansion of 1/sum(n>=1, x^(n - 1)/(2*n - 1)!! ).

1, -1, 2, -2, -2, 2, 46, -46, 314, 194102, -3229166, -663382, 2836767994, -11441854, -3736651874, 2414923738478, 236418596900006, -6139787306, -28607438174617066, 130216032333763994, -621533718480306419638, -93413975428728344583902, 270176365029669324474442

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

G.f.: 1/sum(n>=1, x^(n - 1)/(2*n - 1)!! ) = sqrt(2/Pi) * sqrt(x))/ (exp(x/2) * erf(sqrt(x)/sqrt(2)).

Example

1/(1 + x/3 + x^2/15 +x^3/105 +x^4/945 +x^5/10395 +...) = 1 -x/3 +2x^2/45 -2x^3/945 -2x^4/14175 +2x^5/93555 +46x^6/638512875 -46x^7/273648375 + ...

Mathematica

q[x_] = (Sqrt[2/Pi]*Sqrt[x])/ (E^(x/2)*Erf[Sqrt[x]/Sqrt[2]]) ;
Numerator[CoefficientList[Series[q[x], {x, 0, 30}], x]]

Prog

(PARI) lista(n) = { n++; x = z + z*O(z^n); P = 1/sum(m=1, n, (x^(m - 1)/prod(k=1, m, 2*k-1))); n--; for (i=0, n, print1(numerator(polcoeff(P, i, z)), ", " ); ); } \\ Michel Marcus, Oct 03 2013
(Sage)
def A154288_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (2*k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0].numerator())
    return R
print(A154288_list(23)) # Peter Luschny, Feb 21 2016

Crossrefs

Cf. A154289 (denominators).

Sequence in context: A286851 A320017 A289087 * A225057 A084954 A226281

Adjacent sequences: A154285 A154286 A154287 * A154289 A154290 A154291

Keyword

sign,frac

Author

Roger L. Bagula, Jan 06 2009

Extensions

Edited by Michel Marcus, Oct 03 2013

Status

approved