login
The OEIS is supported by the many generous donors to the OEIS 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 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A338408 E.g.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0. 2
1, 3, 70, 4515, 567576, 116389295, 35111089728, 14574226069095, 7944376570503040, 5494208894263886139, 4694820247236686649600, 4853712224007783889422923, 5968210130160831707746406400, 8605241830169634366425696447655, 14375558607944255605507888571539456 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Compare to: [x^n] (1 + n*x - W(x))^n = 0, for n>0, where W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n! = 1 + x/LambertW(-x).

Compare to: [x^n] (1 + n*x - C(x))^(n+1) = 0, for n>0, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

LINKS

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

FORMULA

a(n) ~ c * d^n * n!^2 / n^2, where d = (1+r) / ((-1 + exp(r + LambertW(-1, -exp(-r)*r))) * LambertW(-exp(-1-r)*(1+r))) = 8.406107401279769476199925123910168..., r = 0.7545302104650497245839827141610818561001159135034... is the root of the equation r*(1 + r + LambertW(-exp(-1 - r)*(1 + r))) = -(1 + r)*(r + LambertW(-1, -exp(-r)*r)) and c = 0.031468237083... - Vaclav Kotesovec, Aug 12 2021, updated Dec 29 2021

EXAMPLE

E.g.f.: A(x) = x + 3*x^2/2! + 70*x^3/3! + 4515*x^4/4! + 567576*x^5/5! + 116389295*x^6/6! + 35111089728*x^7/7! + 14574226069095*x^8/8! + 7944376570503040*x^9/9! + 5494208894263886139*x^10/10! + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in (1 + n*x - A(x))^(2*n) begins:

n=0: [1, 0, 0, 0, 0, 0, 0, 0, ...];

n=1: [1, 0, -6, -140, -8976, -1130952, -232274240, -70128541380, ...];

n=2: [1, 4, 0, -364, -21504, -2530284, -504753152, -149907313980, ...];

n=3: [1, 12, 102, 0, -45960, -5063916, -928551600, -263868802728, ...];

n=4: [1, 24, 480, 7000, 0, -9924168, -1748523008, -457324971720, ...];

n=5: [1, 40, 1410, 42140, 939360, 0, -3259331360, -836926230780, ...];

n=6: [1, 60, 3264, 158220, 6595584, 208807788, 0, -1509806731620, ...];

n=7: [1, 84, 6510, 460936, 29355816, 1626947196, 69489455728, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0.

PROG

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

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A338328, A337758, A350366.

Sequence in context: A082942 A000282 A322775 * A277413 A210920 A140048

Adjacent sequences: A338405 A338406 A338407 * A338409 A338410 A338411

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 24 2020

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)