login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A305940 Irregular triangle where T(n,k) is the coefficient of s(y) in p(n), where s is Schur functions, p is power-sum symmetric functions, and y is the integer partition with Heinz number A215366(n,k). 4
1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, -1, 1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

Row n contains n nonzero entries, half of which (rounded up) are 1 and the remainder are -1.

LINKS

Table of n, a(n) for n=1..81.

FORMULA

T(n,k) = (-1)^(n - A061395(k)) if A215366(n,k) belongs to A093641 (Heinz numbers of hooks), 0 otherwise.

EXAMPLE

Triangle begins:

  1

  1 -1

  1 -1  1

  1  0 -1  1 -1

  1 -1  0  0  1 -1  1

  1  0 -1  0  0  1  0  0 -1  1 -1

  1 -1  0  0  0  1  0  0  0 -1  0  0  1 -1  1

  1 -1  0  0  1  0  0  0  0  0  0  0 -1  0  0  0  1  0  0 -1  1 -1

The rows correspond to the following symmetric function identities.

  p(1) = s(1)

  p(2) = s(2) - s(11)

  p(3) = s(3) - s(21) + s(111)

  p(4) = s(4) - s(31) + s(211) - s(1111)

  p(5) = s(5) - s(41) + s(311) - s(2111) + s(11111)

MATHEMATICA

hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n], {2, _}], {}|{{_, 1}}];

Table[If[hookQ[k], (-1)^(n-Max[PrimePi/@FactorInteger[k][[All, 1]]]), 0], {n, 8}, {k, Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

CROSSREFS

Cf. A056239, A082733, A093641, A153452, A153734, A215366, A296188, A296561.

Cf. A304438, A305940, A317552, A317553, A317554.

Sequence in context: A118172 A071039 A118111 * A119981 A115789 A212793

Adjacent sequences:  A305937 A305938 A305939 * A305941 A305942 A305943

KEYWORD

sign,tabf

AUTHOR

Gus Wiseman, Sep 14 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 01:19 EST 2019. Contains 329978 sequences. (Running on oeis4.)