login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A304400 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) * (2 - A(x)) = 0 for n > 0. 5
1, 1, 8, 153, 4736, 205125, 11606832, 826208992, 72258829312, 7635270104361, 961709587281200, 142709474491679777, 24684776053129473408, 4928830965337886481836, 1126011129156595573835552, 291967631033958376653342600, 85304359600279978669204291584, 27900684466477404020849587348577 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:

(1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,

(2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.

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

A304857(n) = a(n) / n^2 for n >= 1.

LINKS

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

FORMULA

a(n) ~ c * n!^2 * n^2, where c = 0.777184293541721432034108670879422244... - Vaclav Kotesovec, Oct 06 2020

EXAMPLE

O.g.f.: A(x) = 1 + x + 8*x^2 + 153*x^3 + 4736*x^4 + 205125*x^5 + 11606832*x^6 + 826208992*x^7 + 72258829312*x^8 + 7635270104361*x^9 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) * (2 - A(x)) begins:

n=0: [1, -1, -16, -918, -113664, -24615000, -8356919040, ...];

n=1: [1, 0, -15, -920, -113955, -24650904, -8363901035, ...];

n=2: [1, 3, 0, -830, -113088, -24636696, -8363675648, ...];

n=3: [1, 8, 65, 0, -97923, -23962896, -8273887803, ...];

n=4: [1, 15, 240, 3850, 0, -19894104, -7851595520, ...];

n=5: [1, 24, 609, 16432, 444861, 0, -6241325915, ...];

n=6: [1, 35, 1280, 49410, 2034240, 84952296, 0, ...];

n=7: [1, 48, 2385, 123640, 6775197, 399396504, 24384667957, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2 * x*A(x) ) * (2 - 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, 3850/7, 444861/9, 84952296/11, 24384667957/13, ...] = [1, 1, 13, 550, 49429, 7722936, 1875743689, ...].

RELATED SERIES.

exp( x*A(x) ) = 1 + x + 3*x^2/2! + 55*x^3/3! + 3889*x^4/4! + 588201*x^5/5! + 151295251*x^6/6! + 59575340623*x^7/7! + 33795420271425*x^8/8! + ...

Note that the factorial series

F(x) = 1 + x + 2!*x^2 + 3!*x^3 + 4!*x^4 + 5!*x^5 + ... + n!*x^n + ...

satisfies [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n > 0.

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) ) * (2 - Ser(A)) )[m] ); A[n+1]}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A304401, A304402, A304857, A305116.

Sequence in context: A094059 A171202 A247538 * A174845 A302259 A113268

Adjacent sequences:  A304397 A304398 A304399 * A304401 A304402 A304403

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 25 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 04:52 EST 2021. Contains 349562 sequences. (Running on oeis4.)