OFFSET
0,5
COMMENTS
Compare with A377250.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..499
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1/A(-x*A(x)).
(1.b) A(x) = (-1/x) * Series_Reversion(-x*A(x)).
(1.c) A(x) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(n+1) / (n+1).
(1.d) A(x)^m = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(n+m) * m/(n+m) for |m| > 0,
(1.e) A(x) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^n / n ).
(2.a) [x^(2*n+1)] A(x)^n = 0 for n >= 0,
(2.b) [x^(2*n)] 1/A(x)^n = 0 for n >= 1,
(2.c) [x^(2*n+1)] 1/A(x)^(3*n+1) = 0 for n >= 1,
(2.d) [x^(2*n+2)] A(x)^n = -[x^(2*n+3)] A(x)^n for n >= 0 (conjecture),
(2.e) [x^(2*n+2*k+1)] (1+x)^k * A(x)^n = 0 for n >= 0, k >= 0 (conjecture).
A related power series B(x) may be defined by:
(3.a) A(x) = B( x/A(x) ),
(3.b) A(x) = 1 / B( -x*A(x)^2 ),
(3.c) B(x) = A( x*B(x) ),
(3.d) B(x) = 1 / A( -x*B(x)^2 ),
(3.e) B(x) = (1/x) * Series_Reversion(x/A(x)),
(3.f) B(x) = ( (-1/x) * Series_Reversion(-x*A(x)^2) )^(1/2),
(3.g) B(x) = Sum_{n>=0} x^n * [x^n] A(x)^(n+1) / (n+1),
(3.h) B(x) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(2*n+1) / (2*n+1),
(3.i) B(x) = exp( Sum_{n>=1} x^n * [x^n] A(x)^n / n ),
(3.j) B(x) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^(2*n) / (2*n) ).
A related power series C(x) may be defined by:
(4.a) A(x) = C( x/A(x)^2 ),
(4.b) A(x) = 1 / C( -x*A(x)^3 ),
(4.c) C(x) = A( x*C(x)^2 ),
(4.d) C(x) = 1 / A( -x*C(x)^3 ),
(4.e) C(x) = ( (1/x) * Series_Reversion(x/A(x)^2) )^(1/2),
(4.f) C(x) = ( (-1/x) * Series_Reversion(-x*A(x)^3) )^(1/3),
(4.g) C(x) = Sum_{n>=0} x^n * [x^n] A(x)^(2*n+1) / (2*n+1),
(4.h) C(x) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^(3*n+1) / (3*n+1),
(4.i) C(x) = exp( Sum_{n>=1} x^n * [x^n] A(x)^(2*n) / (2*n) ),
(4.j) C(x) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^(3*n) / (3*n) ).
For fixed integer k, there exists a power series F(x,k) that satisfies:
(5.a) A(x) = F( x/A(x)^k, k),
(5.b) A(x) = 1 / F( -x*A(x)^(k+1), k),
(5.c) F(x,k) = A( x*F(x,k)^k ),
(5.d) F(x,k) = 1 / A( -x*F(x,k)^(k+1) ),
(5.e) F(x,k) = ( (1/x) * Series_Reversion(x/A(x)^k) )^(1/k),
(5.f) F(x,k) = ( (-1/x) * Series_Reversion(-x*A(x)^(k+1)) )^(1/(k+1)),
(5.g) F(x,k) = Sum_{n>=0} x^n * [x^n] A(x)^(k*n+1) / (k*n+1),
(5.h) F(x,k) = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^((k+1)*n+1) / ((k+1)*n+1),
(5.i) F(x,k)^m = Sum_{n>=0} x^n * [x^n] A(x)^(k*n+m) * m/(k*n+m) for |m| > 0,
(5.j) F(x,k)^m = Sum_{n>=0} (-x)^n * [x^n] 1/A(x)^((k+1)*n+m) * m/((k+1)*n+m) for |m| > 0,
(5.k) F(x,k) = exp( Sum_{n>=1} x^n * [x^n] A(x)^(k*n) / (k*n) ),
(5.l) F(x,k) = exp( Sum_{n>=1} (-x)^n * [x^n] 1/A(x)^((k+1)*n) / ((k+1)*n) ).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 - 2*x^4 + 2*x^5 + 21*x^6 - 44*x^7 - 435*x^8 + 1416*x^9 + 14642*x^10 - 65280*x^11 - 726906*x^12 + ...
RELATED SERIES.
A related power series B(x) = A(x*B(x)) begins
B(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 13*x^5 + 42*x^6 + 132*x^7 - 1196*x^9 + 8084*x^10 + 75488*x^11 + ...
where A(x) = B(x/A(x)) and A(x) = 1/B(-x*A(x)^2).
B(x)^2 = (-1/x)*Series_Reversion(-x*A(x)^2) = 1 + 2*x + 5*x^2 + 12*x^3 + 26*x^4 + 56*x^5 + 154*x^6 + 456*x^7 + ...
where B(x) = 1/A(-x*B(x)^2).
A related power series C(x) = A(x*C(x)^2) begins
C(x) = 1 + x + 3*x^2 + 11*x^3 + 44*x^4 + 189*x^5 + 868*x^6 + 4165*x^7 + 20247*x^8 + 98972*x^9 + 498810*x^10 + ...
where A(x) = C(x/A(x)^2) and A(x) = 1/C(-x*A(x)^3).
C(x)^2 = (1/x)*Series_Reversion(x/A(x)^2) = 1 + 2*x + 7*x^2 + 28*x^3 + 119*x^4 + 532*x^5 + 2499*x^6 + 12168*x^7 + ...
C(x)^3 = (-1/x)*Series_Reversion(-x*A(x)^3) = 1 + 3*x + 12*x^2 + 52*x^3 + 234*x^4 + 1089*x^5 + 5250*x^6 + 26028*x^7 + ...
where C(x) = 1/A(-x*C(x)^3).
RELATED TABLES.
The table of coefficients of x^k in A(x)^n begins
A^1: [1, 1, 1, 0, -2, 2, 21, -44, -435, 1416, 14642, ...];
A^2: [1, 2, 3, 2, -3, 0, 42, -42, -912, 1866, 31166, ...];
A^3: [1, 3, 6, 7, 0, -3, 58, 0, -1362, 1362, 48054, ...];
A^4: [1, 4, 10, 16, 11, 0, 70, 72, -1731, 0, 63904, ...];
A^5: [1, 5, 15, 30, 35, 21, 90, 170, -1980,-2065, 77516, 0, ...];
A^6: [1, 6, 21, 50, 78, 78, 147, 312, -2070,-4636, 87945, 90312, -4839504, 0, ...]; ...
in which zeros are found at [x^(2*n+1)] A(x)^n for n >= 0.
The table of coefficients of x^k in 1/A(x)^n begins
1/A^1: [1, -1, 0, 1, 1, -6, -14, 87, 318, -2291, ...];
1/A^2: [1, -2, 1, 2, 0, -14, -15, 204, 451, -5258, ...];
1/A^3: [1, -3, 3, 2, -3, -21, 0, 330, 351, -8606, ...];
1/A^4: [1, -4, 6, 0, -7, -24, 30, 440, 0, -11972, ...];
1/A^5: [1, -5, 10, -5, -10, -21, 70, 510, -585, -14965, 0, ...];
1/A^6: [1, -6, 15, -14, -9, -12, 112, 522,-1353, -17212, 18765, 749550, 0, ...];
1/A^7: [1, -7, 21, -28, 0, 0, 147, 468,-2226, -18403, 42861, 851592, -900816, -51997687, 0, ...]; ...
in which zeros are found at [x^(2*n)] 1/A(x)^n for n >= 1.
Zeros are also located at [x^(2*n+1)] 1/A(x)^(3*n+1) for n >= 1.
Notice that the main diagonal of this table equals (-1)^n*(n+1)*a(n):
[1, -2, 3, 0, -10, -12, 147, ...] = [1, -2*(1), 3*(1), -4*(0), 5*(-2), -6*(2), 7*(21), ...];
that is, a(n) = (-1)^n * [x^n] 1/A(x)^(n+1)/(n+1) for n >= 0.
PROG
(PARI) \\ Using [x^(2*n+1)] A(x)^n = 0 and [x^(2*n)] 1/A(x)^n = 0
{a(n) = my(A=[1, 1]); for(m=2, n, A=concat(A, 0);
A[#A] = (1/((-1)^m*(m\2)))*polcoeff( Ser(A)^(-(-1)^m*(m\2)), m); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 24 2024
STATUS
approved