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

%I A026944 #45 Aug 11 2025 01:51:11
%S A026944 1,2,28,1016,69904,7796768,1282366912,291885678464,87844207042816,
%T A026944 33775227494400512,16152024497964817408,9402833148376976193536,
%U A026944 6546848699382209957269504,5372168190357763804164005888,5130820073307731596716765724672,5642704273822755928641583754215424
%N A026944 E.g.f. is inverse function to y -> integral from 0 to y of exp(-s^2).
%C A026944 The generating function is odd, so this list contains only the nonzero coefficients in the Taylor expansion.
%C A026944 Limit_{n->oo} (a(n)/(n!)^2)^(1/n) = 16/Pi. - _Vaclav Kotesovec_, Nov 19 2014
%H A026944 Vaclav Kotesovec, <a href="/A026944/b026944.txt">Table of n, a(n) for n = 1..220</a>
%H A026944 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], 2006-2007.
%H A026944 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/0501052 [math.CA], 2005.
%F A026944 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.
%F A026944 E.g.f.: (1/2*sqrt(Pi)*erf)^{-1}(x).
%F A026944 a(n) = A002067(n-1) * 2^(n-1).
%F A026944 E.g.f. A(x) satisfies the differential equation A'(x) = exp(A(x)^2). - _Vladimir Kruchinin_, Jan 22 2011
%F A026944 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 A026944 MakeTable[n_] := Select[CoefficientList[Series[InverseErf[2x/Sqrt[Pi]],{x,0,2n+1}],x] Table[k!, {k,0,2n+1}], # != 0 &]; MakeTable[15] (* _Emanuele Munarini_, Dec 17 2012 *)
%t A026944 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 A026944 (PARI) v=Vec(serlaplace(serreverse(intformal(exp(-x^2)))));
%o A026944 vector(#v\2,n,v[2*n-1])  /* show terms */
%o A026944 /* Demonstration of Kruchinin's differential equation: */
%o A026944 default(seriesprecision,55); /* that many terms */
%o A026944 A=serreverse(intformal(exp(-x^2))); /* e.g.f. */
%o A026944 deriv(A)-exp(A^2)  /* gives O(x^57), i.e., zero up to order */
%o A026944 (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 A026944 Cf. A002067.
%K A026944 nonn
%O A026944 1,2
%A A026944 _F. Chapoton_, Mar 22 2000