A393763 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+3)*x - n*A(x) ) = 0 for n >= 1.

4, 36, 2604, 471456, 153941040, 77949764208, 56150061828336, 54404955135078912, 68156337346386179040, 107174418868091062967040, 206698301747199115385571072, 479871405497148373579406192640, 1320497324108367814170304802767872, 4251174638430062338734634870654520064, 15833848575753418838990864636144026356480

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) * n^(2*n - 3/2) / (sqrt(2*Pi) * exp(2*n) * (3-w)^(2*n-1) * w^(n+1)), where w = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Mar 02 2026

Example

O.g.f.: A(x) = 4*x + 36*x^2 + 2604*x^3 + 471456*x^4 + 153941040*x^5 + 77949764208*x^6 + 56150061828336*x^7 + 54404955135078912*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*(n+3)*x - n*A(x) ) begin
  n = 1: [1,   0,   -72,  -15624, -11299392, ...];
  n = 2: [1,  12,     0,  -34704, -24171264, ...];
  n = 3: [1,  42,  1548,       0, -40853808, ...];
  n = 4: [1,  96,  8928,  739296,         0, ...];
  n = 5: [1, 180, 32040, 5559480, 867343680, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, equals all zeros.
RELATED SERIES.
exp(A(x)) = 1 + 4*x + 88*x^2/2! + 16552*x^3/3!+ 11587648*x^4/4! + 18713330944*x^5/5! + 56584856838016*x^6/6! + ...
where [x^n] exp(n^2*(n+3)*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(PARI) {a(n, k=3) = 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, 3), ", "))

Crossrefs

Cf. A393760, A317347, A393761, A393762, A393764, A393765, A393766, A393767.

Sequence in context: A374390 A175493 A179870 * A001152 A349848 A384959

Adjacent sequences: A393760 A393761 A393762 * A393764 A393765 A393766

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2026

Status

approved