A302978 Chromatic invariant of the n-path complement graph.

1, 2, -1, 0, 1, 5, 28, 182, 1350, 11274, 104856, 1076064, 12086760, 147561000, 1946180160, 27582811920, 418131670320, 6751768862160, 115707685968000, 2097617321963520, 40108223054148480, 806736096940072320, 17028466642305216000, 376369075678811040000

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Chromatic Invariant

Eric Weisstein's World of Mathematics, Path Complement Graph

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(-1)^k*(n-k-2)! for n > 3. - Andrew Howroyd, Apr 22 2018

a(n) = ((2*n + 3)*a(n + 1) - (n + 5)*a(n + 2) + a(n + 3))/n for n > 2. - Eric W. Weisstein, Apr 23 2018

a(n) ~ exp(-1) * n! / n^2. - Vaclav Kotesovec, Apr 23 2018

Mathematica

Join[{1, 2}, Table[Sum[Binomial[n - k, k] (-1)^k (n - k - 2)!, {k, 0, Floor[n/2]}], {n, 3, 20}]]
Join[{1, 2}, Table[Gamma[n - 1] HypergeometricPFQ[{1/2 - n/2, -n/2}, {2 - n, -n}, -4], {n, 3, 20}]]
Join[{1, 2}, RecurrenceTable[{-n a[n] + (3 + 2 n) a[1 + n] + (-5 - n) a[2 + n] + a[3 + n] == 0, a[1] == -1, a[2] == 0, a[3] == 1}, a, {n, 20}]]

Prog

(PARI) a(n)={if(n<4, [1, 2, -1][n], sum(k=0, n\2, binomial(n-k, k)*(-1)^k*(n-k-2)!))} \\ Andrew Howroyd, Apr 22 2018

Crossrefs

Sequence in context: A275514 A180782 A213597 * A108723 A291584 A352451

Adjacent sequences: A302975 A302976 A302977 * A302979 A302980 A302981

Keyword

sign

Author

Eric W. Weisstein, Apr 16 2018

Extensions

Terms a(17) and beyond from Andrew Howroyd, Apr 22 2018

Status

approved