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E.g.f. is inverse function to y -> integral from 0 to y of exp(-s^2).
3

%I #28 Nov 01 2024 02:03:02

%S 1,2,28,1016,69904,7796768,1282366912,291885678464,87844207042816,

%T 33775227494400512,16152024497964817408,9402833148376976193536,

%U 6546848699382209957269504,5372168190357763804164005888,5130820073307731596716765724672,5642704273822755928641583754215424

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

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

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

%C Limit n->infinity (a(n)/(n!)^2)^(1/n) = 16/Pi. - _Vaclav Kotesovec_, Nov 19 2014

%H Vaclav Kotesovec, <a href="/A026944/b026944.txt">Table of n, a(n) for n = 1..220</a>

%H D. Dominici, <a href="http://arxiv.org/abs/math/0607230">Asymptotic analysis of the derivatives of the inverse error function</a>, arXiv:math/0607230 [math.CA]

%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions.</a> arXiv:math/0501052v2 [math.CA]

%F 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)

%F E.g.f. A(x) satisfies the differential equation A'(x)=exp(A(x)^2) [From _Vladimir Kruchinin_, Jan 22 2011]

%F 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

%t 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]

%t 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 *)

%o (PARI) v=Vec(serlaplace(serreverse(intformal(exp(-x^2)))));

%o vector(#v\2,n,v[2*n-1]) /* show terms */

%o /* Demonstration of Kruchinin's differential equation: */

%o default(seriesprecision,55); /* that many terms */

%o A=serreverse(intformal(exp(-x^2))); /* e.g.f. */

%o deriv(A)-exp(A^2) /* gives O(x^57), i.e., zero up to order */

%o (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 */

%Y Cf. A002067.

%K nonn,changed

%O 1,2

%A _F. Chapoton_, Mar 22 2000