A351918 G.f. A(x) satisfies: [x^n] (1 + n*x - A(x))^(2*n+1) = 0, for n > 0.

1, 2, 16, 262, 6688, 231938, 10130888, 532099310, 32582983264, 2275724862530, 178357712565096, 15489332978290006, 1475661504819976992, 152969444613431797250, 17137064243138924753480, 2062951616302994096028894

Offset: 1

Author

Paul D. Hanna, Feb 25 2022

Keywords

• nonn

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

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).

Examples

			G.f.: A(x) = x + 2*x^2 + 16*x^3 + 262*x^4 + 6688*x^5 + 231938*x^6 + 10130888*x^7 + 532099310*x^8 + 32582983264*x^9 + 2275724862530*x^10 + ...
Related table.
Here we illustrate the formula [x^n] (1 + n*x - A(x))^(2*n+1) = 0 (n > 0).
The table of coefficients of x^k in (1 + n*x - A(x))^(2*n+1) begins:
n=0: [1, -1,   -2,   -16,   -262,   -6688,  -231938, ...];
n=1: [1,  0,   -6,   -48,   -774,  -19872,  -691910, ...];
n=2: [1,  5,    0,  -110,  -1645,  -38439, -1286640, ...];
n=3: [1, 14,   70,     0,  -3374,  -74928, -2272326, ...];
n=4: [1, 27,  306,  1692,      0, -144342, -4262946, ...];
n=5: [1, 44,  858,  9504,  58938,       0, -7887286, ...];
n=6: [1, 65, 1924, 33982, 388401, 2694523,        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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..330

Crossrefs

Cf. A338408, A337758.

Programs

PARI

{a(n) = my(A=[1], m=1); for(i=1, n, A=concat(A, 0); m=#A;
A[m] = polcoeff( (1 + m*x - x*Ser(A))^(2*m+1), m)/(2*m+1) ); A[n]}
for(n=1, 20, print1(a(n), ", "))

Formula

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 = 0.053954066... - Vaclav Kotesovec, Feb 27 2022