A338408 E.g.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n) = 0, for n > 0.

1, 3, 70, 4515, 567576, 116389295, 35111089728, 14574226069095, 7944376570503040, 5494208894263886139, 4694820247236686649600, 4853712224007783889422923, 5968210130160831707746406400, 8605241830169634366425696447655, 14375558607944255605507888571539456

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