A305140 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^2 dx ) / A(x), for n > 0.

1, 1, 33, 3422, 710395, 245288190, 127281447538, 92967363233586, 91202509214139831, 115939599286159123295, 185623891076803234259504, 365706330842590993758556662, 869715472542563657980211015186, 2456766458611829222907737567821138, 8131203421875726862447708824758000364

Offset

0,3

Comments

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^2 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^3 is a g.f. of A001764.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..250

Formula

a(n) ~ 3^(3*n - 1) * n^(2*n - 1/2) / (sqrt(2*Pi*(1-c)) * c^n * (3-c)^(2*n - 1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 19 2020

Example

O.g.f.: A(x) = 1 + x + 33*x^2 + 3422*x^3 + 710395*x^4 + 245288190*x^5 + 127281447538*x^6 + 92967363233586*x^7 + 91202509214139831*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x)^2 dx)/A(x) begins:
n=0: [1, -1, -32, -3357, -702560, -243654950, ...];
n=1: [1, 0, -63/2, -3367, -5633901/8, -2440775421/10, ...];
n=2: [1, 7, 0, -3325, -715316, -1235155194/5, ...];
n=3: [1, 26, 665/2, 0, -5720533/8, -5095053859/20, ...];
n=4: [1, 63, 2016, 41699, 0, -1290302622/5, ...];
n=5: [1, 124, 15561/2, 328643, 80013395/8, 0, ...];
n=6: [1, 215, 23296, 1697283, 93264388, 19574613422/5, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 67*x^2 + 6910*x^3 + 1428723*x^4 + 492223022*x^5 + 255112067610*x^6 + 186210340326168*x^7 + 182601537143712727*x^8 + ...
exp( Integral A(x)^2 dx) = 1 + x + 5*x^2/2! + 415*x^3/3! + 167521*x^4/4! + 172296341*x^5/5! + 355443416701*x^6/6! + 1288266047868955*x^7/7! + 7518341623369166465*x^8/8! + ...
A'(x)/A(x) = 1 + 65*x + 10168*x^2 + 2825845*x^3 + 1222346736*x^4 + 762046826846*x^5 + 649809039848130*x^6 + 728835192043655757*x^7 + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*intformal(Ser(A)^2)) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A305137, A305138, A305139, A305141, A305142, A305143.

Sequence in context: A284072 A281444 A242019 * A282374 A281933 A232292

Adjacent sequences: A305137 A305138 A305139 * A305141 A305142 A305143

Keyword

nonn

Author

Paul D. Hanna, May 31 2018

Status

approved