A383559 O.g.f. A(x) satisfies: [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.

1, 3, 29, 609, 20857, 997671, 61114409, 4548317073, 397323349505, 39774233809179, 4483232458612245, 561425116837715457, 77289022946177141161, 11597365849594347661839, 1883429636306366952452433, 329083700898584984268782241, 61549497773760817234065857793, 12268604214374346472111552473267

Offset: 0

Author

Paul D. Hanna, May 17 2025

Keywords

• nonn

Examples

			O.g.f.: A(x) = 1 + 3*x + 29*x^2 + 609*x^3 + 20857*x^4 + 997671*x^5 + 61114409*x^6 + 4548317073*x^7 + 397323349505*x^8 + ...
RELATED TABLE.
The table of coefficients of x^k/k! in exp( n*(2*n+1)*x ) / A(x) begins
  n = 1: [1,  0,  -49,  -3186,  -445203, -109403892, -41045026725, ...];
  n = 2: [1,  7,    0,  -3872,  -546416, -126698400, -45990717440, ...];
  n = 3: [1, 18,  275,      0,  -664875, -160762914, -55439093817, ...];
  n = 4: [1, 33, 1040,  27900,        0, -196031664, -71849477952, ...];
  n = 5: [1, 52, 2655, 129778,  5408749,          0, -87799444565, ...];
  n = 6: [1, 75, 5576, 407664, 28585872, 1710760608,            0, ...];
  ...
illustrating [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A304319, A337458.

Programs

PARI

{a(n) = my(A=[1],m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*(m-1)*(2*m-1) +x*O(x^m)) / Ser(A) )[m] ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

a(n) ~ sqrt(1-w) * 2^(3*n - 1/4) * n^(n - 1/2) / (sqrt(Pi) * exp(n) * (2-w)^n * w^(n + 1/4)), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, May 18 2025