A320417 O.g.f. A(x) satisfies: [x^n] exp(-n^2*A(x)) / (1 - n^2*x) = 0, for n > 0.

1, 2, 27, 1312, 125725, 19877634, 4644661441, 1501087818944, 640786440035745, 349236672544961550, 236695639072681655042, 195322914258394193939808, 192869728403705883411146031, 224593016480452799339762161070, 304623945406240486265488269648600, 476130992607087098886173799883802624, 849656108159062192953462986972010725625

Offset

1,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

a(n) ~ c * n^(2*n - 2), where c = exp(exp(-2) - 3) * (exp(2) - 1) = 0.36419050799963000048040121372730789359398... - Vaclav Kotesovec, Aug 11 2021, updated Mar 18 2024

Example

O.g.f.: A(x) = x + 2*x^2 + 27*x^3 + 1312*x^4 + 125725*x^5 + 19877634*x^6 + 4644661441*x^7 + 1501087818944*x^8 + 640786440035745*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n^2*A(x)) / (1 - n^2*x) begins:
n=1: [1, 0, -3, -160, -31455, -15082176, -14310224075, ...];
n=2: [1, 0, 0, -520, -124416, -60323424, -57244390400, ...];
n=3: [1, 0, 45, 0, -237951, -134365824, -128906646075, ...];
n=4: [1, 0, 192, 5600, 0, -205474176, -226875814400, ...];
n=5: [1, 0, 525, 27200, 2383425, 0, -306673758875, ...];
n=6: [1, 0, 1152, 87480, 12925440, 1915825824, 0, ...];
n=7: [1, 0, 2205, 227360, 47631969, 11053430976, 2730401653525, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 5*x^2/2! + 175*x^3/3! + 32209*x^4/4! + 15252821*x^5/5! + 14405086381*x^6/6! + 23511056196475*x^7/7! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 151*x^3/3! - 30815*x^4/4! - 14924901*x^5/5! - 14219731019*x^6/6! - 23307795465907*x^7/7! + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^2*x*Ser(A))/(1-m^2*x +x^2*O(x^m))^1)[m+1]/m^2 ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A319938, A319939, A320418.

Sequence in context: A221535 A067075 A015217 * A113094 A327128 A320669

Adjacent sequences: A320414 A320415 A320416 * A320418 A320419 A320420

Keyword

nonn

Author

Paul D. Hanna, Oct 15 2018

Status

approved