A026944 E.g.f. is inverse function to y -> integral from 0 to y of exp(-s^2).

1, 2, 28, 1016, 69904, 7796768, 1282366912, 291885678464, 87844207042816, 33775227494400512, 16152024497964817408, 9402833148376976193536, 6546848699382209957269504, 5372168190357763804164005888, 5130820073307731596716765724672, 5642704273822755928641583754215424

Offset

1,2

Comments

The generating function is odd, so this list contains only the nonzero coefficients in the Taylor expansion.

a(n) = A002067(n) * 2^{n-1}.

Limit n->infinity (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

D. Dominici, Asymptotic analysis of the derivatives of the inverse error function, arXiv:math/0607230 [math.CA]

D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]

Formula

Nonzero constant terms of the polynomials 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)

E.g.f. A(x) satisfies the differential equation A'(x)=exp(A(x)^2) [From 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[2x/Sqrt[Pi]], {x, 0, 2n+1}], x] Table[k!, {k, 0, 2n+1}], # != 0 &] MakeTable[60] [Emanuele Munarini, Dec 17 2012]
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 *)

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

Cf. A002067.

Sequence in context: A012756 A362587 A009403 * A296464 A292806 A113633

Adjacent sequences: A026941 A026942 A026943 * A026945 A026946 A026947

Keyword

nonn

Author

F. Chapoton, Mar 22 2000

Status

approved