A393883 - OEIS

A393883

O.g.f. A(x) satisfies: [x^n] exp( n^8*x - n^3*A(x) ) = 0 for n >= 1.

%I #9 May 18 2026 06:48:13

%S 1,3844,1696840596,11843421298181888,600567913457231735114000,

%T 143229672214318101538770307845696,

%U 121508290214370553374787406992716549713808,301933033756295921904657612229325808594540831678464,1905132525799732817123296403829577016004628248111568229689440

%N O.g.f. A(x) satisfies: [x^n] exp( n^8*x - n^3*A(x) ) = 0 for n >= 1.

%H Paul D. Hanna, <a href="/A393883/b393883.txt">Table of n, a(n) for n = 1..201</a>

%F a(n) ~ sqrt(1-c) * 8^(8*n - 3) * n^(7*n - 7/2) / (sqrt(2*Pi) * (8-c)^(7*n - 3) * c^n * exp(7*n)), where c = -LambertW(-8*exp(-8)). - _Vaclav Kotesovec_, May 18 2026

%e O.g.f.: A(x) = x + 3844*x^2 + 1696840596*x^3 + 11843421298181888*x^4 + 600567913457231735114000*x^5 + 143229672214318101538770307845696*x^6 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k/k! in exp( n^8*x - n^3*A(x) ) begins

%e n = 1: [1, 0, -7688, -10181043576, ...];

%e n = 2: [1, 248, 0, -81478854592, ...];

%e n = 3: [1, 6534, 42485580, 0, ...];

%e n = 4: [1, 65472, 4286090752, 279902918287872, 0, ...];

%e n = 5: [1, 390500, 152489289000, 59545044183053000, 23214878976398803336000, 0, ...]; ...

%e in which a diagonal, the coefficient of x^n in row n, is all zeros.

%e RELATED SERIES.

%e exp(A(x)) = 1 + x + 7689*x^2/2! + 10181066641*x^3/3! + 284242152057901777*x^4/4! + 72068151036861185322722241*x^5/5! + 103125364426751750402197988522653081*x^6/6! + ...

%e where [x^n] exp(n^8*x) / exp(A(x))^(n^3) = 0 for n >= 1.

%o (PARI) {a(n,p=8,q=3) = my(A=[0], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^p*x - m^q*Ser(A)) )[m+1]/(m^q) ); polcoef( Ser(A), n)}

%o for(n=1, 15, print1(a(n,8,3), ", "))

%Y Below, sequences with the parameters (p,q) indicated have an o.g.f. F(x) satisfying [x^n] exp( n^p*x - n^q*F(x) ) = 0 for n >= 1.

%Y Cf. A393881 (8,1), A393882 (8,2), A393884 (8,4), A393885 (8,5), A393886 (8,6), A393887 (8,7).

%Y Cf. A319943 (4,3), A393853 (5,3), A393863 (6,3), A393873 (7,3).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 02 2026