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EXAMPLE
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O.g.f.: A(x) = 1 + 2*x + 14*x^2 + 195*x^3 + 4352*x^4 + 140995*x^5 + 6253116*x^6 + 364374941*x^7 + 27036713728*x^8 + 2491365948867*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 2 - A(x)) begins:
n=0: [1, -2, -28, -1170, -104448, -16919400, -4502243520, ...];
n=1: [2, 0, -22, -1090, -100566, -16477948, -4412412190, ...];
n=2: [3, 4, 0, -738, -84304, -14795544, -4094683392, ...];
n=3: [4, 10, 44, 0, -52212, -11661618, -3524135580, ...];
n=4: [5, 18, 116, 1262, 0, -6827560, -2672928448, ...];
n=5: [6, 28, 222, 3210, 77582, 0, -1509882570, ...];
n=6: [7, 40, 368, 6030, 186864, 9166632, 0, ...];
n=7: [8, 54, 560, 9932, 335376, 21078266, 1896079208, 0, ...]; ....
in which the main diagonal is all zeros after the initial term, illustrating the property that [x^n] exp( n * x*A(x) ) * (n + 2 - A(x)) = 0 for n > 0.
RELATED SERIES.
exp(x*A(x)) = 1 + x + 5*x^2/2! + 97*x^3/3! + 5089*x^4/4! + 550121*x^5/5! + 105094981*x^6/6! + 32297760265*x^7/7! + 14961684191777*x^8/8! + ...
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