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%I #19 Oct 06 2020 03:48:25
%S 1,1,6,69,1292,35795,1381662,71137787,4723263400,393582311685,
%T 40255365257750,4961198428428121,725404352867194164,
%U 124184747391389562735,24610467350470767598014,5590555396845759790005015,1443198598348337621150240912,420171257194869987686583296425,137030366263275866352884688085734,49758716350139864002666237593244717,20008715354387819486054019707449804060
%N O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 1 - A(x)) = 0 for n >= 0.
%C Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
%C (1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
%C (2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%H Paul D. Hanna, <a href="/A305110/b305110.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ c * n!^2, where c = 3.73613614872360588359758608879980516876... - _Vaclav Kotesovec_, Oct 06 2020
%e O.g.f.: A(x) = 1 + x + 6*x^2 + 69*x^3 + 1292*x^4 + 35795*x^5 + 1381662*x^6 + 71137787*x^7 + 4723263400*x^8 + 393582311685*x^9 + 40255365257750*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 1 - A(x)) begins:
%e n=0: [0, -1, -12, -414, -31008, -4295400, -994796640, ...];
%e n=1: [1, 0, -11, -416, -31227, -4312664, -996840335, ...];
%e n=2: [2, 3, 0, -302, -27216, -3971736, -942218944, ...];
%e n=3: [3, 8, 27, 0, -17481, -3201216, -823857561, ...];
%e n=4: [4, 15, 76, 586, 0, -1913384, -633716000, ...];
%e n=5: [5, 24, 153, 1576, 27897, 0, -362610715, ...];
%e n=6: [6, 35, 264, 3114, 69648, 2672616, 0, ...];
%e n=7: [7, 48, 415, 5368, 129579, 6269224, 466272571, 0, ...]; ...
%e in which the coefficients of x^n in row n form a diagonal of zeros.
%e RELATED SERIES.
%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1825*x^4/4! + 164481*x^5/5! + 26795491*x^6/6! + 7156366603*x^7/7! + 2926639231713*x^8/8! + ...
%o (PARI) {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * (m - Ser(A)) )[m] );A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A305111, A305112, A305113, A305114.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 25 2018
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