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%I #13 Mar 02 2026 03:48:21
%S 2,16,1020,165536,49320160,23135212080,15630527512432,
%T 14347969660399360,17168404551237910944,25958066829841770846080,
%U 48399006022125586628932736,109114846124680246152236983296,292658282423729795503719053388032,921152122780387427929905530363699968,3362986922125901260144227986553038280960
%N O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+1)*x - n*A(x) ) = 0 for n >= 1.
%H Paul D. Hanna, <a href="/A393761/b393761.txt">Table of n, a(n) for n = 1..301</a>
%F a(n) ~ sqrt(1-w) * 3^(3*n - 2/3) * n^(2*n - 3/2) / (sqrt(2*Pi) * exp(2*n) * (3-w)^(2*n-1) * w^(n + 1/3)), where w = -A226750 = -LambertW(-3*exp(-3)). - _Vaclav Kotesovec_, Mar 02 2026
%e O.g.f.: A(x) = 2*x + 16*x^2 + 1020*x^3 + 165536*x^4 + 49320160*x^5 + 23135212080*x^6 + 15630527512432*x^7 + 14347969660399360*x^8 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( n^2*(n+1)*x - n*A(x) ) begin
%e n = 1: [1, 0, -32, -6120, -3969792, ...];
%e n = 2: [1, 8, 0, -13264, -8345600, ...];
%e n = 3: [1, 30, 804, 0, -13802544, ...];
%e n = 4: [1, 72, 5056, 321120, 0, ...];
%e n = 5: [1, 140, 19440, 2646200, 328420480, 0, ...];
%e ...
%e in which a diagonal, the coefficient of x^n in row n, equals all zeros.
%e RELATED SERIES.
%e exp(A(x)) = 1 + 2*x + 36*x^2/2! + 6320*x^3/3! + 4025680*x^4/4! + 5960384352*x^5/5! + 16730918782144*x^6/6! + ...
%e where [x^n] exp(n^2*(n+1)*x) / exp(A(x))^n = 0 for n >= 1.
%o (PARI) {a(n,k=1) = my(A=[0], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*(m+k)*x - m*Ser(A)) )[m+1]/m ); polcoef( Ser(A), n)}
%o for(n=1, 15, print1(a(n,1), ", "))
%Y Cf. A393760, A317347, A393762, A393763, A393764, A393765, A393766, A393767.
%K nonn
%O 1,1
%A _Paul D. Hanna_, Mar 01 2026