A393765 O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+5)*x - n*A(x) ) = 0 for n >= 1.

6, 64, 5292, 1072352, 385322400, 211762028976, 163677620352176, 168566053117057280, 222690093537453283872, 366839454924410928031360, 737047142820591826780788352, 1774249919609139600797173509120, 5042273999764573347948713660863744, 16707837561420261525962058483694662400, 63863968134053858316341183877764827956480

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 + 5/3)), where w = -A226750 = -LambertW(-3*exp(-3)). - Vaclav Kotesovec, Mar 02 2026

Example

O.g.f.: A(x) = 6*x + 64*x^2 + 5292*x^3 + 1072352*x^4 + 385322400*x^5 + 211762028976*x^6 + 163677620352176*x^7 + 168566053117057280*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*(n+5)*x - n*A(x) ) begin
  n = 1: [1,   0,  -128,   -31752,  -25687296, ...];
  n = 2: [1,  16,     0,   -71696,  -55668224, ...];
  n = 3: [1,  54,  2532,        0,  -95557680, ...];
  n = 4: [1, 120, 13888,  1416672,          0, ...];
  n = 5: [1, 220, 47760, 10066840, 1889541760, 0, ...];
  ...
in which a diagonal, the coefficient of x^n in row n, equals all zeros.
RELATED SERIES.
exp(A(x)) = 1 + 6*x + 164*x^2/2! + 34272*x^3/3! + 26576592*x^4/4! + 47064613536*x^5/5! + 154208308022976*x^6/6! + ...
where [x^n] exp(n^2*(n+5)*x) / exp(A(x))^n = 0 for n >= 1.

Prog

(PARI) {a(n, k=5) = 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, 5), ", "))

Crossrefs

Cf. A393760, A317347, A393761, A393762, A393763, A393764, A393766, A393767.

Sequence in context: A249592 A333983 A087488 * A249896 A249828 A179879

Adjacent sequences: A393762 A393763 A393764 * A393766 A393767 A393768

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2026

Status

approved