A304402 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.

1, 1, 9, 179, 5661, 249424, 14337039, 1035838044, 91867414241, 9833503227827, 1253246430314670, 187948018130914066, 32818034910964227439, 6608081830970361618546, 1520982783352578794866344, 397027611766464517915252056, 116698001659938095895315068553, 38375694701199964362412343063161

Offset

0,3

Comments

Note: [x^n] exp( n * x*G(x) ) / G(x) = 0 for n>0 when G(x) is the g.f. of A088716.

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

What is the limit A304402(n) / A304400(n) ? Seems to be near 1.51...

A304402(n) / A304400(n) tends to 1.522998920075488836991600223419379... - Vaclav Kotesovec, Oct 06 2020

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * n!^2 * n^2, where c = 1.18365083976367345437640389636650727... - Vaclav Kotesovec, Oct 06 2020

Example

O.g.f.: A(x) = 1 + x + 9*x^2 + 179*x^3 + 5661*x^4 + 249424*x^5 + 14337039*x^6 + 1035838044*x^7 + 91867414241*x^8 + 9833503227827*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) / A(x) begins:
n=0: [1, -1, -16, -972, -125952, -28275000, -9885939840, ...];
n=1: [1, 0, -15, -968, -125835, -28263864, -9883855835, ...];
n=2: [1, 3, 0, -860, -123456, -28073976, -9850185728, ...];
n=3: [1, 8, 65, 0, -104811, -26970576, -9680119083, ...];
n=4: [1, 15, 240, 3892, 0, -21937464, -9078485120, ...];
n=5: [1, 24, 609, 16528, 457173, 0, -7077136715, ...];
n=6: [1, 35, 1280, 49572, 2066880, 89033736, 0, ...];
n=7: [1, 48, 2385, 123880, 6839349, 411165624, 26124539077, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.
Terms along the secondary diagonal in the above table are divisible by the odd numbers: [1, 3/3, 65/5, 3892/7, 457173/9, 89033736/11, 26124539077/13, ...] = [1, 1, 13, 556, 50797, 8093976, 2009579929, ...].
RELATED SERIES.
exp( x*A(x) ) = 1 + x + 3*x^2/2! + 61*x^3/3! + 4537*x^4/4! + 702501*x^5/5! + 183891571*x^6/6! + 73567995313*x^7/7! + 42361186187601*x^8/8! + ...
The arithmetic inverse of the o.g.f. begins:
1/A(x) = 1 - x - 8*x^2 - 162*x^3 - 5248*x^4 - 235625*x^5 - 13730472*x^6 - 1001798042*x^7 - 89479215104*x^8 - 9627430506669*x^9 + ...

Prog

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

Crossrefs

Cf. A304400, A088716.

Sequence in context: A003711 A009009 A220267 * A358741 A034221 A034240

Adjacent sequences: A304399 A304400 A304401 * A304403 A304404 A304405

Keyword

nonn

Author

Paul D. Hanna, May 25 2018

Status

approved