A305139 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^2 dx ) / A(x), for n > 0.

1, 1, 9, 155, 3805, 118632, 4429279, 191275884, 9340355265, 507681357635, 30360217294454, 1979895257720082, 139811654124752231, 10629630950986800850, 865954337592580081080, 75286721276048241037848, 6961094538227014053702537, 682423909436661488354778945, 70743106543492192940195723155, 7736186700358670253328879658965

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..300

Formula

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.652748826558... - Vaclav Kotesovec, Oct 19 2020

Example

O.g.f.: A(x) = 1 + x + 9*x^2 + 155*x^3 + 3805*x^4 + 118632*x^5 + 4429279*x^6 + 191275884*x^7 + 9340355265*x^8 + 507681357635*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x)^2 dx)/A(x) begins:
n=0: [1, -1, -8, -138, -3440, -108905, -4118952, -179740162, ...];
n=1: [1, 0, -15/2, -140, -28065/8, -111009, -67094895/16, -1279321635/7, ...];
n=2: [1, 3, 0, -130, -3660, -117117, -4419200, -1344147030/7, ...];
n=3: [1, 8, 65/2, 0, -27265/8, -124886, -76650687/16, -2911952885/14, ...];
n=4: [1, 15, 120, 630, 0, -117081, -5202600, -1614205230/7, ...];
n=5: [1, 24, 609/2, 2712, 139455/8, 0, -78693087/16, -3562210803/14, ...];
n=6: [1, 35, 640, 8190, 81940, 620323, 0, -1698895510/7, ...];
n=7: [1, 48, 2385/2, 20540, 2215455/8, 3088737, 428675377/16, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^2 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 19*x^2 + 328*x^3 + 8001*x^4 + 247664*x^5 + 9188337*x^6 + 394725252*x^7 + 19194243265*x^8 + 1039762257722*x^9 + ...
exp( Integral A(x)^2 dx) = 1 + x + 3*x^2/2! + 45*x^3/3! + 2145*x^4/4! + 203085*x^5/5! + 30980475*x^6/6! + 6838973145*x^7/7! + 2045481775425*x^8/8! + 792696897387225*x^9/9! + ...
A'(x)/A(x) = 1 + 17*x + 439*x^2 + 14473*x^3 + 568296*x^4 + 25625759*x^5 + 1297831032*x^6 + 72732570537*x^7 + 4462331350255*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^2*intformal(Ser(A)^2)) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A365015 A208998 A236578 * A208545 A183471 A109677

Adjacent sequences: A305136 A305137 A305138 * A305140 A305141 A305142

Keyword

nonn

Author

Paul D. Hanna, May 31 2018

Status

approved