%I #7 Oct 27 2018 11:13:13

%S 1,2,6,32,220,1812,17108,180512,2093760,26396160,358741328,5223336288,

%T 81079811280,1336407320080,23311138957200,429063111959808,

%U 8311760620707648,169072470759431232,3603666131945918144,80327823251439861760,1869212211081119135616,45331401566332423284864,1143967734536203174726784,29996686272924492809481216,816185909551276017516640000

%N O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x - n*x^2) = 0, for n > 0.

%C It is remarkable that this sequence should consist entirely of integers.

%C Compare to: [x^n] exp(n*G(x)) * (1 - n*x) = 0, for n > 0, when G(x) = x + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A088716.

%F O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x).

%e O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 220*x^5 + 1812*x^6 + 17108*x^7 + 180512*x^8 + 2093760*x^9 + 26396160*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x - n*x^2) begins:

%e n=1: [1, 0, 1, 28, 729, 26416, 1321225, 87466716, ...];

%e n=2: [1, 0, 0, 32, 1200, 49152, 2569600, 172974720, ...];

%e n=3: [1, 0, -3, 0, 1089, 60408, 3509325, 246760776, ...];

%e n=4: [1, 0, -8, -80, 0, 49024, 3777280, 293683968, ...];

%e n=5: [1, 0, -15, -220, -2535, 0, 2848825, 291386100, ...];

%e n=6: [1, 0, -24, -432, -7056, -105984, 0, 208089216, ...];

%e n=7: [1, 0, -35, -728, -14175, -293048, -5733875, 0, ...];

%e n=8: [1, 0, -48, -1120, -24576, -590592, -15603200, -391709184, 0, ...]; ...

%e in which the coefficient of x^n in row n forms a diagonal of zeros.

%e RELATED SERIES.

%e (a) Differential Equation.

%e O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x) where

%e A'(x) = 1 + 4*x + 18*x^2 + 128*x^3 + 1100*x^4 + 10872*x^5 + 119756*x^6 + ...

%e A(x)*A'(x) = x + 6*x^2 + 32*x^3 + 220*x^4 + 1812*x^5 + 17108*x^6 + 17108*x^7 + ...

%e so that A(x) - x*A(x)*A'(x) = x + x^2.

%e (b) Exponentiation.

%e exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 985*x^4/4! + 32321*x^5/5! + 1544701*x^6/6! + 99637105*x^7/7! + 8257877489*x^8/8! + ...

%e exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 599*x^4/4! - 21681*x^5/5! - 1106939*x^6/6! - 74873737*x^7/7! - 6431021295*x^8/8! + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m*x*Ser(A))*(1-m*x-m*x^2 +x^2*O(x^m)))[m+1]/m ); A[n]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A088716, A321085, A321087.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 27 2018