A305145 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral 1/A(x) dx ) / A(x), for n > 0.

1, 1, 21, 1886, 381735, 134584434, 72514796422, 55192152857400, 56287911330435339, 74043167807482274450, 122040226074154110294114, 246341047594913378800486668, 597752265070243363135031803950, 1716967839431601765698468898047292, 5762431350664488199395983555754160140, 22346478647255335081326815815314403748524

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..198

Formula

a(n) ~ sqrt(1-c) * 3^(3*n - 1) * n^(2*n - 1/2) / (sqrt(2*Pi) * 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 + 21*x^2 + 1886*x^3 + 381735*x^4 + 134584434*x^5 + 72514796422*x^6 + 55192152857400*x^7 + 56287911330435339*x^8 + 74043167807482274450*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral 1/A(x) dx)/A(x) begins:
n=0: [1, -1, -20, -1845, -377584, -133748650, -72227419704, -55040493806445, ...];
n=1: [1, 0, -21, -1872, -379890, -134201604, -72383437035, -55123034324112, ...];
n=2: [1, 7, 0, -2033, -396970, -137452068, -73490534208, -55705843833995, ...];
n=3: [1, 26, 304, 0, -437155, -147006370, -76635381186, -57333497856168, ...];
n=4: [1, 63, 1932, 36075, 0, -163035066, -83375170872, -60709861617885, ...];
n=5: [1, 124, 7605, 304780, 8444291, 0, -92858506104, -66905102463320, ...];
n=6: [1, 215, 22984, 1625463, 84879650, 3287781224, 0, -74725745263095, ...];
n=7: [1, 342, 58290, 6597132, 556856100, 37129859844, 1920530286186, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3 * Integral 1/A(x) dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - x - 20*x^2 - 1845*x^3 - 377584*x^4 - 133748650*x^5 - 72227419704*x^6 - 55040493806445*x^7 - 56174066916766400*x^8 - 73928074251625193826*x^9 + ...
exp( Integral 1/A(x) dx) = 1 + x - 7*x^3 - 468*x^4 - 75978*x^5 - 22366934*x^6 - 10340491005*x^7 - 6890379290514*x^8 - 6248442860989378*x^9 - 7399048902607246248*x^10 + ..., which is an integer series.
A'(x)/A(x) = 1 + 41*x + 5596*x^2 + 1518597*x^3 + 670826996*x^4 + 434225271374*x^5 + 385813724342292*x^6 + 449847594913097949*x^7 + 665870324595294969196*x^8 + ...

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(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, A305146, A305147, A304861, A304862.

Sequence in context: A297652 A243664 A232949 * A296686 A163928 A358803

Adjacent sequences: A305142 A305143 A305144 * A305146 A305147 A305148

Keyword

nonn

Author

Paul D. Hanna, May 31 2018

Status

approved