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!)
A319944 O.g.f. A(x) satisfies: [x^n] exp( n^5*x - n*A(x) ) = 0  for n >= 1. 5
1, 225, 714000, 10430111250, 455589570897000, 46993311212615010000, 9839324906977709480400000, 3761494651833327732316790250000, 2427487105139453587868600367048750000, 2489491831933123075592260875312720412500000, 3867129529486594159007141093572270035942600000000 (list; graph; refs; listen; history; text; internal format)
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) ~ sqrt(1-c) * 5^(5*n - 1) * n^(4*n - 3/2) / (sqrt(2*Pi) * c^n * (5-c)^(4*n - 1) * exp(4*n)), where c = -LambertW(-5*exp(-5)) = 0.03488576825572369630124086867... - Vaclav Kotesovec, Oct 13 2020

EXAMPLE

G.f.: A(x) = x + 225*x^2 + 714000*x^3 + 10430111250*x^4 + 455589570897000*x^5 + 46993311212615010000*x^6 + 9839324906977709480400000*x^7 + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp( n^5*x - n*A(x) ) begins:

n=1: [1, 0, -450, -4284000, -250322062500, ...];

n=2: [1, 30, 0, -8622000, -501675120000, ...];

n=3: [1, 240, 56250, 0, -760449262500, ...];

n=4: [1, 1020, 1038600, 1038564000, 0, ...];

n=5: [1, 3120, 9732150, 30328848000, 93108209197500, 0, ...];

n=6: [1, 7770, 60370200, 469008792000, 3641608218960000, 27906215370093360000, 0, ...]; ...

in which the coefficient of x^n in row n forms a diagonal of zeros.

RELATED SERIES.

exp(A(x)) = 1 + x + 451*x^2/2! + 4285351*x^3/3! + 250340416201*x^4/4! + 54672019444872001*x^5/5! + 33835513974650405264251*x^6/6! + ...

The 5th root of A(x)/x appears to be an integer sequence:

(A(x)/x)^(1/5) = 1 + 45*x + 138750*x^2 + 2060865000*x^3 + 90706765441275*x^4 + 9381160956625666875*x^5 + 1966116273013953349582500*x^6 + 751938952953001936098785681250*x^7 + 485360862323214790797483583171389375*x^8 + 497810555195750107907248882311441377821875*x^9 + ...

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^5*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}

for(n=1, 15, print1(a(n), ", "))

CROSSREFS

Sequence in context: A304314 A109688 A195277 * A013757 A151653 A077729

Adjacent sequences:  A319941 A319942 A319943 * A319945 A319946 A319947

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 02 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 3 16:36 EST 2021. Contains 349467 sequences. (Running on oeis4.)