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

%I A383559 #11 May 24 2025 03:06:05
%S A383559 1,3,29,609,20857,997671,61114409,4548317073,397323349505,
%T A383559 39774233809179,4483232458612245,561425116837715457,
%U A383559 77289022946177141161,11597365849594347661839,1883429636306366952452433,329083700898584984268782241,61549497773760817234065857793,12268604214374346472111552473267
%N A383559 O.g.f. A(x) satisfies: [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.
%H A383559 Paul D. Hanna, <a href="/A383559/b383559.txt">Table of n, a(n) for n = 0..400</a>
%F A383559 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
%e A383559 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 + ...
%e A383559 RELATED TABLE.
%e A383559 The table of coefficients of x^k/k! in exp( n*(2*n+1)*x ) / A(x) begins
%e A383559   n = 1: [1,  0,  -49,  -3186,  -445203, -109403892, -41045026725, ...];
%e A383559   n = 2: [1,  7,    0,  -3872,  -546416, -126698400, -45990717440, ...];
%e A383559   n = 3: [1, 18,  275,      0,  -664875, -160762914, -55439093817, ...];
%e A383559   n = 4: [1, 33, 1040,  27900,        0, -196031664, -71849477952, ...];
%e A383559   n = 5: [1, 52, 2655, 129778,  5408749,          0, -87799444565, ...];
%e A383559   n = 6: [1, 75, 5576, 407664, 28585872, 1710760608,            0, ...];
%e A383559   ...
%e A383559 illustrating [x^n] exp( n*(2*n+1)*x ) / A(x) = 0 for n > 0.
%o A383559 (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]}
%o A383559 for(n=0, 25, print1( a(n), ", "))
%Y A383559 Cf. A304319, A337458.
%K A383559 nonn
%O A383559 0,2
%A A383559 _Paul D. Hanna_, May 17 2025