A393761 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+1)*x - n*A(x) ) = 0 for n >= 1.

2, 16, 1020, 165536, 49320160, 23135212080, 15630527512432, 14347969660399360, 17168404551237910944, 25958066829841770846080, 48399006022125586628932736, 109114846124680246152236983296, 292658282423729795503719053388032, 921152122780387427929905530363699968, 3362986922125901260144227986553038280960

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..301

Formula

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

Example

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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*(n+1)*x - n*A(x) ) begin
  n = 1: [1,   0,   -32,   -6120,  -3969792, ...];
  n = 2: [1,   8,     0,  -13264,  -8345600, ...];
  n = 3: [1,  30,   804,       0, -13802544, ...];
  n = 4: [1,  72,  5056,  321120,         0, ...];
  n = 5: [1, 140, 19440, 2646200, 328420480, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, equals all zeros.
RELATED SERIES.
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! + ...
where [x^n] exp(n^2*(n+1)*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(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)}
for(n=1, 15, print1(a(n, 1), ", "))

Crossrefs

Cf. A393760, A317347, A393762, A393763, A393764, A393765, A393766, A393767.

Sequence in context: A084595 A273193 A002543 * A125791 A102103 A337070

Adjacent sequences: A393758 A393759 A393760 * A393762 A393763 A393764

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2026

Status

approved