A032347 Inverse binomial transform of A032346.

1, 0, 1, 2, 6, 21, 82, 354, 1671, 8536, 46814, 273907, 1700828, 11158746, 77057021, 558234902, 4230337018, 33448622893, 275322101318, 2354401779494, 20878592918183, 191682453823420, 1819147694792802

Offset: 0

Author

Joe K. Crump (joecr(AT)carolina.rr.com)

Keywords

• nonn
• nice
• easy
• eigen

Links

• Branko Dragovich, On Summation of p-Adic Series, arXiv:1702.02569 [math.NT], 2017.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A032346, A046934.

Programs

Mathematica

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 *)
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 *)

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)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Jul 10 2020

Formula

E.g.f. satisfies A' = exp(x) A - 1.

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

G.f. A(x) satisfies: A(x) = 1 - x * (1 - A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, Jul 10 2020