A195588 a(n) = Sum_{k=0..2*n} (-1)^k * moebius(2*n-k+1) * moebius(k+1).

1, -3, -1, 2, 3, 1, 1, 6, 0, -3, 5, 2, 1, -1, 5, -8, 5, 10, -7, 2, 3, -9, -1, 6, 4, -3, 8, 2, -3, 3, -11, 2, 13, -15, 5, -2, -5, 5, 13, -8, -7, 9, 3, -2, 18, -1, -7, -4, -14, -6, 7, -4, -3, 2, 1, 6, 7, 7, -9, 18, -13, 7, 14, -12, 1, -7, 1, 0, -3, -13, 7, 6, -3, -5, 22, -16, 3, -1, -11, 2, 8, -5, -15, 6, 1, -9, 3, 18, 1, 10, -13, 8, 9, 3, -15, -2, -7, 6, 16, -4, 1, 1, 3, -2, 49, -7, -9, -6, -1, -9, -3, -20, -13, -11, -11, -22, 12, 25, 7, 0, -6, 5, 3, -2, -18, 4, 7, 4, -1, -7, -5, -2, -15, 3, 32, 2, 15, 11, -1, 12, 5, -23, 3, -2, -17, 1, 10, 4, 7, 16, 13, 34, -2, -31, -11, -12

Offset

0,2

Comments

It is conjectured that all integers appear an infinite number of times.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1001

Formula

G.f. A(x) satisfies: A(x^2) = M(x)*M(-x) where M(x) = Sum_{n>=0} moebius(n+1)*x^n.

G.f. A(x) = exp( Sum_{n>=1} A195589(2*n)*x^n/n ), where A195589 is the unsigned logarithmic derivative of the Moebius function A008683.

Example

G.f.: A(x) = 1 - 3*x - x^2 + 2*x^3 + 3*x^4 + x^5 + x^6 + 6*x^7 +...
where A(x^2) = M(x)*M(-x) and M(x) begins:
M(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 +...+ moebius(n+1)*x^n +...
log(A(x)) = -3*x - 11*x^2 - 30*x^3 - 83*x^4 - 243*x^5 - 710*x^6 - 2061*x^7 - 6099*x^8 +...+ -A195589(2*n)*x^n/n +...
Positions of zeros begin:
[8,67,119,161,167,206,207,243,260,263,271,331,339,350,371,407,543,803,...].
Positions of other values of a(n) begin:
+1: [0,5,6,12,54,64,66,84,88,100,101,145,202,210,256,290,309,318,321,...];
-1: [2,13,22,45,77,108,128,138,165,180,216,229,236,348,389,390,418,...];
+2: [3,11,19,27,31,53,79,135,242,360,362,413,548,800,839,...];
-2: [35,43,95,103,123,131,143,152,159,197,235,251,299,324,337,349,...];
+3: [4,20,29,42,76,86,93,102,122,133,142,201,240,326,333,401,518,585,...];
-3: [1,9,25,28,52,68,72,110,166,196,204,234,253,280,340,432,472,653,...];
+4: [24,125,127,147,170,211,269,278,332,459,807,...];
-4: [47,51,99,168,422,538,599,...];
+5: [10,14,16,34,37,121,140,177,308,382,484,520,537,642,645,706,741,...];
-5: [36,73,81,130,173,186,193,217,232,257,302,312,357,373,444,448,...].

Prog

(PARI) {a(n)=sum(k=0, 2*n, (-1)^k*moebius(2*n-k+1)*moebius(k+1))}
(PARI) {A195589(n)=n*polcoeff(-log(sum(m=0, n, moebius(m+1)*x^m)+x*O(x^n)), n)}
{a(n)=polcoeff(exp(sum(m=1, n, -A195589(2*m)*x^m/m)+x*O(x^n)), n)}

Crossrefs

Cf. A195589, A008683 (Moebius).

Sequence in context: A163746 A004591 A350090 * A153510 A288537 A167373

Adjacent sequences: A195585 A195586 A195587 * A195589 A195590 A195591

Keyword

sign

Author

Paul D. Hanna, Sep 20 2011

Status

approved