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%I #26 Jul 10 2020 19:04:27
%S 1,0,1,2,6,21,82,354,1671,8536,46814,273907,1700828,11158746,77057021,
%T 558234902,4230337018,33448622893,275322101318,2354401779494,
%U 20878592918183,191682453823420,1819147694792802
%N Inverse binomial transform of A032346.
%H Branko Dragovich, <a href="https://arxiv.org/abs/1702.02569">On Summation of p-Adic Series</a>, arXiv:1702.02569 [math.NT], 2017.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F E.g.f. satisfies A' = exp(x) A - 1.
%F Recurrence: a(1)=0, a(2)=1, for n > 2, a(n) = 1 + Sum_{j=2..n-1} binomial(n-1, j)*a(j). - _Jon Perry_, Apr 26 2005
%F G.f. A(x) satisfies: A(x) = 1 - x * (1 - A(x/(1 - x)) / (1 - x)). - _Ilya Gutkovskiy_, Jul 10 2020
%t a[0] = 1; a[1] = 0; a[n_] := a[n] = 1 + Sum[Binomial[n-1, j]*a[j], {j, 2, n-1}]; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Oct 08 2013, after _Jon Perry_ *)
%t nmax = 20; Assuming[x > 0, CoefficientList[Series[E^(E^x) * (1/E + ExpIntegralEi[-1] - ExpIntegralEi[-E^x]), {x, 0, nmax}], x] ] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 10 2020 *)
%o (PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1 - x * (1 - subst(A, x, x/(1-x)) / (1 - x))); polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Jul 10 2020
%Y Cf. A032346, A046934.
%K nonn,nice,easy,eigen
%O 0,4
%A Joe K. Crump (joecr(AT)carolina.rr.com)
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