A393767 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+7)*x - n*A(x) ) = 0 for n >= 1.

8, 100, 9372, 2112608, 832430320, 495809795952, 411274611814192, 450793834408701952, 629361060265493013600, 1089020313832325733363200, 2286424655192980985937664256, 5725724207595852651005393424384, 16862084987493105044015399763728384, 57704796366891042292335094602319176448, 227133234247586475629600925343176561596160

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 + 4/3) * n^(2*n - 3/2) / (sqrt(2*Pi) * exp(2*n) * (3-w)^(2*n-1) * w^(n + 7/3)), where w = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Mar 02 2026

Example

O.g.f.: A(x) = 8*x + 100*x^2 + 9372*x^3 + 2112608*x^4 + 832430320*x^5 + 495809795952*x^6 + 411274611814192*x^7 + 450793834408701952*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*(n+7)*x - n*A(x) ) begin
  n = 1: [1,   0,  -200,   -56232,  -50582592, ...];
  n = 2: [1,  20,     0,  -128464, -110722304, ...];
  n = 3: [1,  66,  3756,        0, -192270384, ...];
  n = 4: [1, 144, 19936,  2415456,          0, ...];
  n = 5: [1, 260, 66600, 16514840, 3621240640, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, equals all zeros.
RELATED SERIES.
exp(A(x)) = 1 + 8*x + 264*x^2/2! + 61544*x^3/3! + 52702912*x^4/4! + 102074051328*x^5/5! + 362016476209024*x^6/6! + ...
where [x^n] exp(n^2*(n+7)*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(PARI) {a(n, k=7) = 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, 7), ", "))

Crossrefs

Cf. A393760, A317347, A393761, A393762, A393763, A393764, A393765, A393766.

Sequence in context: A215875 A317598 A238947 * A291536 A229452 A199816

Adjacent sequences: A393764 A393765 A393766 * A393768 A393769 A393770

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2026

Status

approved