A294812 Let b(n) be the number of permutations {c_1..c_n} of {1..n} for which c_1 - c_2 + ... + (-1)^(n-1)*c_n are pentagonal numbers (A000326). Then a(n) = b(n)/A010551(n).

1, 1, 1, 2, 4, 5, 6, 10, 23, 38, 70, 110, 196, 346, 759, 1250, 2313, 3982, 8433, 14520, 29437, 50466, 102830, 179587, 376439, 654374, 1343540, 2352149, 4916286, 8654120, 18065200, 31783592, 66233160, 117371504, 246610521, 436972949, 913862320, 1626523783

Offset

1,4

Comments

All terms are positive integers (for a proof, cf. comment in A293984).

Note that a(1), a(2), a(3), a(4) remain the same, if in the definition the pentagonal numbers are replaced by k-gonal numbers for k >= 3 other than k=4.

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..200

Mathematica

polyQ[order_, n_]:=If[n==0, True, IntegerQ[(#-4+Sqrt[(#-4)^2+8 n (#-2)])/(2 (#-2))]&[order]]; (*is a number polygonal?*)
Map[Total, Table[
possibleSums=Range[1/2-(-1)^n/2-Floor[n/2]^2, Floor[(n+1)/2]^2];
filteredSums=Select[possibleSums, polyQ[5, #]&&#>-1&];
positions=Map[Flatten[{#, Position[possibleSums, #, 1]-1}]&, filteredSums];
Map[SeriesCoefficient[QBinomial[n, Floor[(n+1)/2], q], {q, 0, #[[2]]/2}]&, positions], {n, 25}]] (* Peter J. C. Moses, Jan 02 2018 *)

Crossrefs

Cf. A010551, A293857, A293984, A294811.

Sequence in context: A091678 A080053 A101964 * A217354 A165701 A129305

Adjacent sequences: A294809 A294810 A294811 * A294813 A294814 A294815

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 09 2017

Status

approved