A319144 E.g.f. A(x) satisfies: [x^(n-1)] exp(n^2*x) / A(x)^n = 0 for n>1.

1, 2, 7, 82, 3413, 310306, 47180827, 10609392242, 3284088709897, 1333647722701378, 686179134994911311, 435599748089861536402, 334122749226062422725277, 304457064400271021354494562, 324970210527067394401358110243, 401523372735670670696974799321266, 568490192646838149936392483264664977, 914248292513326978923735948784457567362

Offset

0,2

Comments

It is remarkable that the logarithmic derivative of the e.g.f. A(x) should be an integer series.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

a(n) ~ sqrt(1-c) * 2^(2*n - 1) * n^(2*n - 1) / (exp(2*n) * c^n * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, Aug 11 2021

Example

E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 82*x^3/3! + 3413*x^4/4! + 310306*x^5/5! + 47180827*x^6 + 10609392242*x^7/7! + 3284088709897*x^8/8! + 1333647722701378*x^9/9! + 686179134994911311*x^10/10! + 435599748089861536402*x^11/11! + 334122749226062422725277*x^12/12! +...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^2*x) / A(x)^n begins
n=1: [1, -1, -2, -48, -2616, -262080, -41718240, -9630270720, ...];
n=2: [1, 0, -6, -112, -5592, -547968, -86345120, -19809990912, ...];
n=3: [1, 3, 0, -222, -10728, -958824, -144971712, -32519314080, ...];
n=4: [1, 8, 52, 0, -18648, -1693248, -236690784, -50727983616, ...];
n=5: [1, 15, 210, 2420, 0, -2739720, -399251600, -80125144800, ...];
n=6: [1, 24, 558, 12192, 221184, 0, -616918320, -131299591680, ...];
n=7: [1, 35, 1204, 40278, 1272768, 33597312, 0, -196436730672, ...];
n=8: [1, 48, 2280, 106688, 4869552, 210771456, 7654459648, 0, ...]; ...
in which the n-th term in row n forms a diagonal of zeros after an initial '1'.
RELATED SERIES.
The logarithmic derivative of the e.g.f. appears to be an integer series:
A'(x)/A(x) = 2 + 3*x + 28*x^2 + 475*x^3 + 11556*x^4 + 362418*x^5 + 13820696*x^6 + 617990499*x^7 + 31613351140*x^8 + 1817581003238*x^9 + ... + A319146(n+1)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(m=1, n+1, A=concat(A, 0); A[m] = Vec( exp(m^2*x +x*O(x^n))/Ser(A)^(m) )[m]/m ); H=A; A[n+1]}
for(n=0, 21, print1(a(n)*n!, ", "))

Crossrefs

Cf. A319146, A317343, A303060.

Sequence in context: A135756 A263368 A208806 * A111822 A062764 A163855

Adjacent sequences: A319141 A319142 A319143 * A319145 A319146 A319147

Keyword

nonn

Author

Paul D. Hanna, Sep 18 2018

Status

approved