OFFSET
1,2
COMMENTS
The generating function is odd, so this list contains only the nonzero coefficients in the Taylor expansion.
Limit_{n->oo} (a(n)/(n!)^2)^(1/n) = 16/Pi. - Vaclav Kotesovec, Nov 19 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..220
L. Carlitz, The inverse of the error function, Pacific J. Math. 13(2), 459-470 (1963).
D. Dominici, Asymptotic analysis of the derivatives of the inverse error function, arXiv:math/0607230 [math.CA], 2006-2007.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, Int. J. Math. Math. Sci. 2003, 457271 (2003). arXiv version, arXiv:math/0501052 [math.CA], 2005.
FORMULA
Nonzero constant terms of the polynomials P_{2n-1} in t defined by P_1=1, P_{n+1} = P'n+2*n*t*P_n.
E.g.f.: (1/2*sqrt(Pi)*erf)^{-1}(x).
a(n) = A002067(n-1) * 2^(n-1).
E.g.f. A(x) satisfies the differential equation A'(x) = exp(A(x)^2). - Vladimir Kruchinin, Jan 22 2011
Let D denote the operator g(x) -> d/dx(exp(x^2)*g(x)). Then a(n) = D^(2*n-2)(1) evaluated at x = 0. See [Dominici, Example 11]. - Peter Bala, Sep 08 2011
MATHEMATICA
MakeTable[n_] := Select[CoefficientList[Series[InverseErf[2 x/Sqrt[Pi]], {x, 0, 2 n + 1}], x] Table[k!, {k, 0, 2 n + 1}], # != 0 &]; MakeTable[15] (* Emanuele Munarini, Dec 17 2012 *)
(* Alternative: *)
nmax = 20; c = ConstantArray[0, nmax]; c[[1]] = 1; Do[c[[k + 1]] = Sum[c[[m + 1]] * c[[k - m]]/(m + 1)/(2*m + 1), {m, 0, k - 1}], {k, 1, nmax - 1}]; A026944 = c * (2 * Range[0, nmax - 1])! (* Vaclav Kotesovec, Feb 25 2014 *)
(* Alternative, this program is not efficient for a large number of terms: *)
nmax = 20; Table[If[m == 0, 1, BellY[Table[{(2 m + k)!, -If[Mod[k, 2] == 0, (-4)^(k/2) (k - 1)!!^2 (k/2)!/((k + 1)!), 0]}, {k, 2 m}]]/((2 m)!)], {m, 0, nmax}] (* Jan Mangaldan, Apr 24 2026 *)
PROG
(PARI) v=Vec(serlaplace(serreverse(intformal(exp(-x^2)))));
vector(#v\2, n, v[2*n-1]) /* show terms */
/* Demonstration of Kruchinin's differential equation: */
default(seriesprecision, 55); /* that many terms */
A=serreverse(intformal(exp(-x^2))); /* e.g.f. */
deriv(A)-exp(A^2) /* gives O(x^57), i.e., zero up to order */
(Maxima) f(n):=n!/2*coeff(taylor(2*inverse_erf(2*x/sqrt(%pi)), x, 0, n), x, n); makelist(f(2*n+1), n, 0, 12); /* Emanuele Munarini, Dec 17 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, Mar 22 2000
STATUS
approved