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A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).

%I A345652 #66 Dec 23 2024 22:56:59
%S A345652 1,0,-1,2,0,-16,65,-78,-749,6232,-22068,-28920,1004685,-7408740,
%T A345652 22263215,157632230,-2874256740,21590948480,-53087332675,
%U A345652 -956539294506,16344490525835,-132605481091060,294656170409328,9113173803517344,-167298122286332823
%N A345652 Expansion of the e.g.f. exp(-1 + (x + 1)*exp(-x)).
%C A345652 For all p prime, a(p)/(p-1) == 1 (mod p). - _Mélika Tebni_, Mar 21 2022
%H A345652 Seiichi Manyama, <a href="/A345652/b345652.txt">Table of n, a(n) for n = 0..560</a>
%F A345652 The e.g.f. y(x) satisfies y' = -x*y*exp(-x).
%F A345652 a(n) = Sum_{k=0..n-2} (n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k) for n > 0.
%F A345652 Conjecture: a(n) = 0 for only n = 1 and n = 4.
%F A345652 Conjecture: For all p prime, a(p)^2 == 1 (mod p).
%F A345652 Stronger conjecture: For n > 1, a(n) == -1 (mod n) iff n is a prime or 6. - _M. F. Hasler_, Jun 23 2021
%F A345652 a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*Bell(k)*A106828(n, k). - _Mélika Tebni_, Sep 21 2021
%F A345652 a(n) = Sum_{k=0..n} (-1)^k*A003725(n-k)*Bell(k)*binomial(n, k). - _Mélika Tebni_, Mar 21 2022
%e A345652 exp(-1+(x+1)*exp(-x)) = 1 - x^2/2! + 2*x^3/3! - 16*x^5/5! + 65*x^6/6! - 78*x^7/7! - 749*x^8/8! + 6232*x^9/9! + ...
%p A345652 a := series(exp(-1+(x+1)*exp(-x)), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);
%p A345652 a := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end: seq(a(n), n=0..24);
%p A345652 # third program:
%p A345652 A345652 := n -> add((-1)^(n-k)*combinat[bell](k)*A106828(n, k), k=0..iquo(n, 2)):
%p A345652 seq(A345652(n), n=0..24); # _Mélika Tebni_, Sep 21 2021
%t A345652 nmax = 24; CoefficientList[Series[Exp[-1+(x+1)*Exp[-x]], {x, 0, nmax}], x] Range[0, nmax]!
%o A345652 (PARI) seq(n) = {Vec(serlaplace(exp(-1+(x+1)*exp(-x + O(x*x^n)))))} \\ _Andrew Howroyd_, Jun 21 2021
%o A345652 (PARI) a(n) = if(n==0, 1, sum(k=2, n, (-1)^(k-1)*(k-1)*binomial(n-1, k-1)*a(n-k))); \\ _Seiichi Manyama_, Mar 15 2022
%Y A345652 Cf. A003725, A014182, A327006.
%Y A345652 Cf. A292935 (without 1+x: EGF e^(e^(-x)-1)), A000110 (absolute values: Bell numbers, EGF e^(e^x-1))
%Y A345652 Cf. A003725, A106828, A347210, A347571.
%K A345652 sign
%O A345652 0,4
%A A345652 _Mélika Tebni_, Jun 21 2021