OFFSET
1,2
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).
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 8*x^3 + 41*x^4 + 284*x^5 + 2594*x^6 + 29420*x^7 + 395845*x^8 + 6109724*x^9 + 105714438*x^10 + 2017696504*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1 + n*x + n*x^2 - A(x))^(n+1) begins:
n=0: [1, -1, -3, -8, -41, -284, -2594, -29420, -395845, ...];
n=1: [1, 0, -4, -16, -78, -536, -4960, -57048, -775089, ...];
n=2: [1, 3, 0, -29, -171, -1071, -9124, -100590, -1334241, ...];
n=3: [1, 8, 24, 0, -340, -2504, -19032, -189408, -2368430, ...];
n=4: [1, 15, 95, 290, 0, -5327, -46335, -409770, -4606315, ...];
n=5: [1, 24, 252, 1472, 4614, 0, -103528, -1028952, -10296567, ...];
n=6: [1, 35, 546, 4949, 27972, 90244, 0, -2388773, -26537259, ...];
n=7: [1, 48, 1040, 13376, 112344, 627280, 2083504, 0, -63579020, ...];
n=8: [1, 63, 1809, 31260, 360765, 2901258, 16172964, 55276020, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1 + n*x + n*x^2 - A(x))^(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, 3, 41, 395845, 15928490898945133, 490833755530209698774408313021523960879357, ...].
RELATED SERIES.
B(x) = 1/(1 - A(x)) = 1 + x + 4*x^2 + 15*x^3 + 76*x^4 + 478*x^5 + 3868*x^6 + 39675*x^7 + 498120*x^8 + 7351430*x^9 + 123503516*x^10 + 2309531318*x^11 + ...
where [x^n] (1 + n*(x + x^2)*B(x))^(n+1) / B(x)^(n+1) = 0 for n > 0.
Series_Reversion(A(x)) = x - 3*x^2 + 10*x^3 - 56*x^4 + 268*x^5 - 2104*x^6 + 7636*x^7 - 129976*x^8 - 369988*x^9 - 19147364*x^10 - 279267684*x^11 - ...
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 + m*x^2 - x*Ser(A))^(m+1), m)/(m+1) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 24 2020
STATUS
approved