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%I #9 Feb 28 2022 13:40:50
%S 1,1,1,2,6,22,96,486,2816,18362,133092,1060918,9226068,86913822,
%T 881783456,9584972462,111135773688,1369122271498,17858966209908,
%U 245895213956190,3563864413568516,54235164104218478,864658341720196176,14411626441272698566
%N E.g.f. A(x) satisfies: Sum_{k=0..n} [x^k/k!] 1/A(x)^(n+1-k) = 0 for n > 0.
%H Vaclav Kotesovec, <a href="/A351919/b351919.txt">Table of n, a(n) for n = 0..220</a>
%F a(n) ~ c * d^n * n!, where d = 0.72467075187356681806169214268514... and c = 0.9182939975437585609088613585... - _Vaclav Kotesovec_, Feb 28 2022
%e G.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 6*x^4/4! + 22*x^5/5! + 96*x^6/6! + 486*x^7/7! + 2816*x^8/8! + 18362*x^9/9! + 133092*x^10/10! + ...
%e Related series.
%e 1/A(x) = 1 - x + x^2/2! - 2*x^3/3! + 4*x^4/4! - 12*x^5/5! + 38*x^6/6! - 150*x^7/7! + 648*x^8/8! - 3218*x^9/9! + ... + A213058(n)*x^n + ...
%e log(A(x)) = x + x^3/3! + x^4/4! + 6*x^5/5! + 14*x^6/6! + 86*x^7/7! + 342*x^8/8! + 2394*x^9/9! + 13648*x^10/10! + ...
%e Illustration of definition.
%e The table of coefficients of x^k in A(x)^(-n) for n > 0 begins:
%e n=1: [1, -1, 1, -2, 4, -12, 38, -150, ...];
%e n=2: [1, -2, 4, -10, 30, -104, 420, -1896, ...];
%e n=3: [1, -3, 9, -30, 114, -486, 2316, -12210, ...];
%e n=4: [1, -4, 16, -68, 316, -1608, 8936, -54024, ...];
%e n=5: [1, -5, 25, -130, 720, -4280, 27330, -187230, ...];
%e n=6: [1, -6, 36, -222, 1434, -9792, 70908, -544800, ...];
%e n=7: [1, -7, 49, -350, 2590, -20034, 162680, -1389066, ...];
%e n=8: [1, -8, 64, -520, 4344, -37616, 339216, -3193200, ...]; ...
%e in which the antidiagonals add to zero (after the initial term):
%e 0 = 1 + (-1) ;
%e 0 = 1 + (-2) + 1 ;
%e 0 = 1 + (-3) + 4 + (-2) ;
%e 0 = 1 + (-4) + 9 + (-10) + 4 ;
%e 0 = 1 + (-5) + 16 + (-30) + 30 + (-12) ;
%e 0 = 1 + (-6) + 25 + (-68) + 114 + (-104) + 38 ;
%e ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = sum(k=0,#A-1, k!*polcoeff( 1/Ser(A)^(#A-k) ,k)) / (#A-1)! ); n!*A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A213058.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Feb 25 2022