A042961 The sequence e when b is obtained by reversing the parity of Euler's partition function A000041.

1, 0, 1, 0, 2, 0, 2, 0, 3, 1, 3, 1, 4, 2, 4, 3, 6, 4, 8, 5, 10, 7, 13, 8, 16, 11, 19, 15, 23, 18, 29, 25, 34, 30, 43, 38, 54, 46, 64, 58, 79, 68, 93, 86, 112, 105, 133, 127, 160, 156, 187, 188, 228, 226, 270, 269, 320, 323, 381, 379, 450, 456, 531, 538, 625, 637

Offset

0,5

Comments

Map a binary sequence b=[ b_1,... ] to a binary sequence c=[ c_1,... ] so that C=1/Product (1-x^i)^c_i == 1+Sum b_i*x^i mod 2.

This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C=1+Sum e_i*x^i.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(u=vector(n, i, 1-numbpart(i)%2), v=vector(n)); for(n=1, #v, v[n]=(u[n] + EulerT(v[1..n])[n])%2); concat([1], EulerT(v))} \\ Andrew Howroyd, May 04 2021

Crossrefs

Cf. A042951, A042960, A042962.

Sequence in context: A302643 A319973 A025804 * A029190 A166865 A300575

Adjacent sequences: A042958 A042959 A042960 * A042962 A042963 A042964

Keyword

nonn

Author

N. J. A. Sloane and J. H. Conway

Extensions

Terms a(46) and beyond from Andrew Howroyd, May 04 2021

Status

approved