A338328 G.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n-1) = 0, for n > 0.

1, 1, 8, 129, 3216, 108770, 4638624, 238318885, 14304161568, 981167968494, 75656236536880, 6474624435825546, 608726166485138400, 62351661805733365988, 6910034200942823999216, 823702177681649409615885, 105083183676401369775220288, 14284797856254053619382213974

Offset

1,3

Comments

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).

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).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) is odd iff n is a power of 2 (conjecture).

a(n) = 2 (mod 4) iff n is twice the sum of two distinct powers of 2 (conjecture).

a(n) ~ c * d^n * n! / 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 = 0.0183535737... - Vaclav Kotesovec, Oct 25 2020, updated Dec 29 2021

Example

G.f.: A(x) = x + x^2 + 8*x^3 + 129*x^4 + 3216*x^5 + 108770*x^6 + 4638624*x^7 + 238318885*x^8 + 14304161568*x^9 + 981167968494*x^10 + 75656236536880*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1 + n*x - A(x))^(2*n-1) begins:
n=0: [1, 1, 2, 11, 150, 3522, 116004, 4875971, 248249926, ...];
n=1: [1, 0, -1, -8, -129, -3216, -108770, -4638624, -238318885, ...];
n=2: [1, 3, 0, -29, -435, -10395, -344980, -14551710, -742232889, ...];
n=3: [1, 10, 35, 0, -995, -22108, -685040, -27845120, -1387317400, ...];
n=4: [1, 21, 182, 763, 0, -44352, -1308496, -49463691, -2359476623, ...];
n=5: [1, 36, 567, 5016, 24795, 0, -2460798, -90517248, -4024228311, ...];
n=6: [1, 55, 1364, 19987, 188111, 1076779, 0, -163739444, -7210725280, ...];
n=7: [1, 78, 2795, 60736, 886665, 8986692, 58632756, 0, -12708845682, ...];
n=8: [1, 105, 5130, 154475, 3196890, 47836926, 523400300, 3840855735, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x - A(x))^(2*n-1) = 0, for n > 0.
ODD TERMS.
The odd terms seem to occur only at positions equal to powers of 2: a(1), a(2), a(4), a(8), a(16), ...; the odd terms begin: [1, 1, 129, 238318885, 823702177681649409615885, 1728309826852815676925583353579702599032084743126745454358749, ...].
RELATED SERIES.
1/(1 - A(x)) = 1 + x + 2*x^2 + 11*x^3 + 150*x^4 + 3522*x^5 + 116004*x^6 + 4875971*x^7 + 248249926*x^8 + 14807944838*x^9 + 1011137601996*x^10 + ...
Series_Reversion(A(x)) = x - x^2 - 6*x^3 - 94*x^4 - 2404*x^5 - 83808*x^6 - 3667808*x^7 - 192327976*x^8 - 11726343040*x^9 - 814059155216*x^10 - ...

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-1), m)/(2*m-1) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A337758, A000108, A018900, A350366.

Sequence in context: A348207 A356914 A364986 * A073701 A239756 A295240

Adjacent sequences: A338325 A338326 A338327 * A338329 A338330 A338331

Keyword

nonn

Author

Paul D. Hanna, Oct 24 2020

Status

approved