A305147 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral 1/A(x)^2 dx ) / A(x), for n > 0.

1, 1, 1, 13, 201, 4799, 146509, 5465853, 239779725, 12065090215, 683788505469, 43055465865105, 2979786144976833, 224718173520876855, 18335712354871184749, 1609062791960716840469, 151097465043129176493237, 15116317905498147638860983, 1605008879121294393641990077, 180254723532204767389702764585, 21348717445490413966641543430233

Offset: 0

Author

Paul D. Hanna, May 31 2018

Keywords

• nonn

Comments

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

Note: 0 = [x^n] exp( n * Integral 1/F(x)^2 dx ) / F(x) holds for n > 0 when 1/F(-x) = C(x) = 1 + x*C(x)^2 is a g.f. of A000108.

Examples

			O.g.f.: A(x) = 1 + x + x^2 + 13*x^3 + 201*x^4 + 4799*x^5 + 146509*x^6 + 5465853*x^7 + 239779725*x^8 + 12065090215*x^9 + 683788505469*x^10 + 43055465865105*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral 1/A(x)^2 dx)/A(x) begins:
n=0: [1, -1, 0, -12, -176, -4410, -136968, -5173266, ...];
n=1: [1, 0, -3/2, -12, -1545/8, -23229/5, -2282987/16, -187096983/35, ...];
n=2: [1, 3, 0, -20, -252, -27426/5, -808448/5, -41341014/7, ...];
n=3: [1, 8, 45/2, 0, -3241/8, -37566/5, -16103943/80, -98105421/14, ...];
n=4: [1, 15, 96, 308, 0, -57474/5, -282824, -315815478/35, ...];
n=5: [1, 24, 525/2, 1688, 50967/8, 0, -6694523/16, -179820699/14, ...];
n=6: [1, 35, 576, 5868, 40420, 894366/5, 0, -649016238/35, ...];
n=7: [1, 48, 2205/2, 16060, 1320759/8, 6216189/5, 510096457/80, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2 * Integral 1/A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
1/A(x) = 1 - 2*x + x^2 - 24*x^3 - 328*x^4 - 8468*x^5 - 264972*x^6 - 10068372*x^7 - 447223340*x^8 - 22709482068*x^9 - 1296038603112*x^10 + ...
exp( Integral 1/A(x)^2 dx) = 1 + x - x^2/2! - 3*x^3/3! - 135*x^4/4! - 8571*x^5/5! - 1061361*x^6/6! - 197712639*x^7/7! - 52240421007*x^8/8! - 18481482225495*x^9/9! + ...
A'(x)/A(x) = 1 + x + 37*x^2 + 753*x^3 + 22991*x^4 + 849829*x^5 + 37219617*x^6 + 1873928193*x^7 + 106404715099*x^8 + 6716223979161*x^9 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..344

Crossrefs

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

Cf. A305144, A305145, A305146.

Programs

PARI

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

Formula

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.0311339300124... - Vaclav Kotesovec, Oct 19 2020