A262940 G.f. satisfies: [x^n] A(x)^(2^m) = 2^m, where m = A007814(n+1) is the highest exponent of 2 dividing n+1, for n>=0.

1, 1, 1, -3, 1, 3, 1, 21, 1, -21, 1, 255, 1, -255, 1, 478677, 1, -478677, 1, 7152407, 1, -7152407, 1, -1291535081, 1, 1291535081, 1, -21021866227, 1, 21021866227, 1, 8367123104756933, 1, -8367123104756933, 1, 125486744208053623, 1, -125486744208053623, 1, -22639240870533272321, 1, 22639240870533272321, 1, -368298497943774746859, 1, 368298497943774746859, 1, -1120119534438107659394201, 1, 1120119534438107659394201

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Formula

a(2*n) = 1 for n>=0.

a(4*n+1) = -a(4*n-1) for n>0.

Coefficient of x^k in A(x)^(2^n) equals 2^n at k = m*2^(n+1) + 2^n - 1 for m>=0.

Example

G.f.: A(x) = 1 + x + x^2 - 3*x^3 + x^4 + 3*x^5 + x^6 + 21*x^7 + x^8 - 21*x^9 + x^10 + 255*x^11 + x^12 - 255*x^13 + x^14 + 478677*x^15 + x^16 +...
The coefficients in A(x)^n begin:
n=1: [1, 1,  1,  -3,   1,    3,    1,  21,    1,  -21,    1,   255, ...];
n=2: [1, 2,  3,  -4,  -3,    2,   19,  44,   29,    2, -153,   512, ...];
n=3: [1, 3,  6,  -2,  -9,   -9,   40, 102,  117,   51, -354,   504, ...];
n=4: [1, 4, 10,   4, -13,  -32,   44, 200,  341,  220, -586,     4, ...];
n=5: [1, 5, 15,  15, -10,  -64,   10, 310,  775,  755, -679, -1305, ...];
n=6: [1, 6, 21,  32,   6,  -96,  -78, 372, 1443, 2030,  -45, -3528, ...];
n=7: [1, 7, 28,  56,  42, -112, -224, 302, 2275, 4459, 2520, -5852, ...];
n=8: [1, 8, 36,  88, 106,  -88, -412,   8, 3075, 8352, 8888, -5568, ...]; ...
where the coefficient of x^k in A(x)^(2^m) = 2^m where m = A007814(k+1) for k>=0, like so:
[x^0] A(x)^1 = 1;
[x^1] A(x)^2 = 2;
[x^2] A(x)^1 = 1;
[x^3] A(x)^4 = 4;
[x^4] A(x)^1 = 1;
[x^5] A(x)^2 = 2;
[x^6] A(x)^1 = 1;
[x^7] A(x)^8 = 8; ...

Prog

(PARI) {a(n) = local(A=[1, 1]); for(k=3, n+1, A=concat(A, 0); m=2^valuation(k, 2); A[k] = 1 - Vec(Ser(A)^m)[k]/m ); A[n+1]}
for(n=0, 64, print1(a(n), ", "))

Crossrefs

Cf. A262939.

Sequence in context: A079412 A356655 A306861 * A278601 A281038 A263677

Adjacent sequences: A262937 A262938 A262939 * A262941 A262942 A262943

Keyword

sign,look

Author

Paul D. Hanna, Oct 04 2015

Status

approved