A304438 Coefficient of s(y) in p(|y|), where s is Schur functions, p is power-sum symmetric functions, y is the integer partition with Heinz number n, and |y| = Sum y_i.

0, 1, 1, -1, 1, -1, 1, 1, 0, -1, 1, 1, 1, -1, 0, -1, 1, 0, 1, 1, 0, -1, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, 0, -1, 0, 0, 1, -1, 0, -1, 1, 0, 1, 1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, 0, 1, 1, 0, 0, 1, 0, 1, -1, 0, 1, 0, 0, 1, 1, 0, -1, 1, 0, 0, -1, 0, -1, 1, 0, 0, 1, 0, -1, 0, -1, 1, 0, 0, 0, 1, 0, 1, -1, 0

Offset

1

Comments

a(1) = 0 by convention.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = (-1)^(A056239(n) - A061395(n)) if n belongs to A093641 (Heinz numbers of hooks), 0 otherwise.

Example

Sum_{n > 0} p(n) = s(1) + s(2) - s(11) + s(3) - s(21) + s(4) + s(111) - s(31) + s(5) + s(211) + s(6) - s(41) - s(1111) + s(7) + s(8) + s(311) + ...

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n], {2, _}], {}|{{_, 1}}];
Table[If[hookQ[n], (-1)^(Total[primeMS[n]]-Max[primeMS[n]]), 0], {n, 2, 100}]

Prog

(PARI)
A000265(n) = (n/2^valuation(n, 2));
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A304438(n) = if(1==n, 0, my(o=A000265(n)); if(((o>1)&&!isprime(o)), 0, (-1)^(A056239(n)-A061395(n)))); \\ Antti Karttunen, Sep 30 2018

Crossrefs

Cf. A000085, A056239, A082733, A093641, A124794, A124795, A153452, A296188, A296561, A300121, A305940, A317552, A317553, A317554.

Sequence in context: A377868 A377874 A374128 * A168181 A368916 A324732

Adjacent sequences: A304435 A304436 A304437 * A304439 A304440 A304441

Keyword

sign

Author

Gus Wiseman, Sep 14 2018

Extensions

More terms from Antti Karttunen, Sep 30 2018

Status

approved