A305145 - OEIS

%I #15 Oct 19 2020 08:30:36

%S 1,1,21,1886,381735,134584434,72514796422,55192152857400,

%T 56287911330435339,74043167807482274450,122040226074154110294114,

%U 246341047594913378800486668,597752265070243363135031803950,1716967839431601765698468898047292,5762431350664488199395983555754160140,22346478647255335081326815815314403748524

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

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

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

%H Vaclav Kotesovec, <a href="/A305145/b305145.txt">Table of n, a(n) for n = 0..198</a>

%F 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

%e 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 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in exp(n^3*Integral 1/A(x) dx)/A(x) begins:

%e n=0: [1, -1, -20, -1845, -377584, -133748650, -72227419704, -55040493806445, ...];

%e n=1: [1, 0, -21, -1872, -379890, -134201604, -72383437035, -55123034324112, ...];

%e n=2: [1, 7, 0, -2033, -396970, -137452068, -73490534208, -55705843833995, ...];

%e n=3: [1, 26, 304, 0, -437155, -147006370, -76635381186, -57333497856168, ...];

%e n=4: [1, 63, 1932, 36075, 0, -163035066, -83375170872, -60709861617885, ...];

%e n=5: [1, 124, 7605, 304780, 8444291, 0, -92858506104, -66905102463320, ...];

%e n=6: [1, 215, 22984, 1625463, 84879650, 3287781224, 0, -74725745263095, ...];

%e n=7: [1, 342, 58290, 6597132, 556856100, 37129859844, 1920530286186, 0, ...]; ...

%e 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.

%e RELATED SERIES.

%e 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 + ...

%e 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.

%e 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 + ...

%o (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]}

%o for(n=0, 20, print1(a(n), ", "))

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

%Y Cf. A305144, A305146, A305147, A304861, A304862.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 31 2018