OFFSET
0,3
FORMULA
G.f. A(x) satisfies: A(x)^4 - A(x)^3 = x. - Paul D. Hanna, Nov 27 2014
G.f. A(x) satisfies: [x^n] A(x)^(3*n+k) = 0 for k=1..n-1, n>1.
G.f. A(x) satisfies: [x^n] A(x)^(3*n) = 3*(-1)^(n-1) for n>0.
G.f. A(x) satisfies: [x^n] A(x)^(4*n) = 4 for n>0.
G.f. A(x) satisfies: [x^n] A(x)^(5*n) = 5*C(2n-1,n) for n>0.
G.f. A(x) = 1 + Series_Reversion(x*(1+x)^3).
G.f. A(x) = [x/Series_Reversion(x*(1+x)^3)]^(1/3).
a(n) = 3*(-1)^(n-1)*C(4*n-1,n-1)/(4*n-1) = (-1)^(n-1)*A006632(n) for n>0.
EXAMPLE
G.f.: A(x) = 1 + x - 3*x^2 + 15*x^3 - 91*x^4 + 612*x^5 +...
The coefficients in the successive powers of g.f. A(x) begin:
A^1: [1, 1, -3, 15, -91, 612, -4389, 32890, -254475, ...];
A^2: [1, 2, -5, 24, -143, 952, -6783, 50600, -390195, ...];
A^3: [1, 3, -6, 28, -165, 1092, -7752, 57684, -444015, ...];
A^4: [1, 4, -6, 28, -165, 1092, -7752, 57684, -444015, ...];
A^5: [1, 5, -5, 25, -150, 1001, -7140, 53295, -411125, ...];
A^6: [1, 6, -3, 20, -126, 858, -6188, 46512, -360525, ...];
A^7: [1, 7, 0, 14, -98, 693, -5096, 38760, -302841, ...];
A^8: [1, 8, 4, 8, -70, 528, -4004, 31008, -245157, ...];
A^9: [1, 9, 9, 3, -45, 378, -3003, 23868, -191862, ...];
A^10: [1, 10, 15, 0, -25, 252, -2145, 17680, -145350, ...];
A^11: [1, 11, 22, 0, -11, 154, -1452, 12584, -106590, ...];
A^12: [1, 12, 30, 4, -3, 84, -924, 8580, -75582, ...];
A^13: [1, 13, 39, 13, 0, 39, -546, 5577, -51714, ...];
A^14: [1, 14, 49, 28, 0, 14, -294, 3432, -34034, ...];
A^15: [1, 15, 60, 50, 0, 3, -140, 1980, -21450, ...];
A^16: [1, 16, 72, 80, 4, 0, -56, 1056, -12870, ...];
A^17: [1, 17, 85, 119, 17, 0, -17, 510, -7293, ...];
A^18: [1, 18, 99, 168, 45, 0, -3, 216, -3861, ...];
A^19: [1, 19, 114, 228, 95, 0, 0, 76, -1881, ...];
A^20: [1, 20, 130, 300, 175, 4, 0, 20, -825, ...];
A^21: [1, 21, 147, 385, 294, 21, 0, 3, -315, ...];
A^22: [1, 22, 165, 484, 462, 66, 0, 0, -99, ...];
A^23: [1, 23, 184, 598, 690, 161, 0, 0, -23, ...];
A^24: [1, 24, 204, 728, 990, 336, 4, 0, -3, ...];
...
PROG
(PARI) {a(n)=polcoeff(1+serreverse(x*(1+x+O(x^(n+1)))^3), n)}
(PARI) {a(n)=if(n==0, 1, 3*(-1)^(n-1)*binomial(4*n-1, n-1)/(4*n-1))}
(PARI) /* From A(x)^4 - A(x)^3 = x: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(1+Ser(A)^4-Ser(A)^3)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 23 2010
STATUS
approved