A305146 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^4 * Integral 1/A(x) dx ) / A(x), for n > 0.

1, 1, 105, 71030, 143839875, 639147831054, 5268190256643730, 72401453092661090460, 1539974714406342828684915, 47967103851505667222316762710, 2096230585920937730055252273554166, 124208697361885403106994025647669349700, 9703933918967416448770462097241544278503550, 976615535896268261227542752682139965289070564940

Offset

0,3

Comments

Note: 0 = [x^n] exp( n * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = 1 + x.

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..144

Formula

a(n) ~ sqrt(1-c) * 2^(8*n - 5/2) * n^(3*n - 1/2) / (sqrt(Pi) * exp(3*n) * c^n * (4-c)^(3*n - 1)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647... - Vaclav Kotesovec, Oct 19 2020

Example

O.g.f.: A(x) = 1 + x + 105*x^2 + 71030*x^3 + 143839875*x^4 + 639147831054*x^5 + 5268190256643730*x^6 + 72401453092661090460*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^4*Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -1, -104, -70821, -143687104, -638845480750, -5266877186423376, ...];
n=1: [1, 0, -105, -70960, -143775630, -639017901600, -5267622501808905, ...];
n=2: [1, 15, 0, -72605, -145123140, -641617076562, -5278826440840960, ...];
n=3: [1, 80, 3055, 0, -149843050, -653149632064, -5327910150826725, ...];
n=4: [1, 255, 32280, 2624475, 0, -678395417454, -5464268996914000, ...];
n=5: [1, 624, 194271, 40142304, 6023531646, 0, -5698446198253501, ...];
n=6: [1, 1295, 837760, 360867555, 116236431740, 29089429020014, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^4 * Integral 1/A(x) dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - x - 104*x^2 - 70821*x^3 - 143687104*x^4 - 638845480750*x^5 - 5266877186423376*x^6 - 72390764082089330493*x^7 + ...
exp( Integral 1/A(x) dx) = 1 + x - 35*x^3 - 17740*x^4 - 28755126*x^5 - 106502983600*x^6 - 752517500258415*x^7 - 9049597920124635300*x^8 - 171101127726280225469450*x^9 + ..., which is an integer series.
A'(x)/A(x) = 1 + 209*x + 212776*x^2 + 575053749*x^3 + 3194983074896*x^4 + 31605201852299630*x^5 + 506772757749658101024*x^6 + 12319213675791316095636957*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^4*intformal(1/Ser(A))) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.

Cf. A305144, A305145, A305147.

Sequence in context: A348794 A028917 A145627 * A304644 A171116 A199100

Adjacent sequences: A305143 A305144 A305145 * A305147 A305148 A305149

Keyword

nonn

Author

Paul D. Hanna, May 31 2018

Status

approved