A294735 Triangle read by rows: T(n,k) (3 <= k < n) gives number of solutions to certain 1-loop scattering equations refined by MHV degree.

2, 6, 6, 14, 44, 14, 30, 210, 210, 30, 62, 832, 1812, 832, 62, 126, 2982, 12012, 12012, 2982, 126, 254, 10068, 68322, 124952, 68322, 10068, 254, 510, 32730, 352350, 1065930, 1065930, 352350, 32730, 510, 1022, 103784, 1700456, 7972568, 13103540, 7972568, 1700456, 103784, 1022

Offset

4,1

Comments

See Appendix E of Farrow and Lipstein.

Links

Table of n, a(n) for n=4..48.

Joseph A. Farrow and Arthur E. Lipstein, From 4d Ambitwistor Strings to On Shell Diagrams and Back, arXiv:1705.07087 [hep-th], 2017.

Formula

T(n,k) = euler(n-1,k-1) - euler(n-2,k-1) - euler(n-2,k-2), where euler = A008292.

Example

Triangle begins
2
6, 6
14, 44, 14
30, 210, 210, 30
62, 832, 1812, 832, 62
126, 2982, 12012, 12012, 2982, 126
254, 10068, 68322, 124952, 68322, 10068, 254
510, 32730, 352350, 1065930, 1065930, 352350, 32730, 510
1022, 103784, 1700456, 7972568, 13103540, 7972568, 1700456, 103784, 1022

Mathematica

e[n_, k_] := Sum[(-1)^j (k - j)^n Binomial[n + 1, j], {j, 0, k}];
T[n_, k_] := e[n - 1, k - 1] - e[n - 2, k - 1] - e[n - 2, k - 2];
Table[T[n, k], {n, 4, 12}, {k, 3, n - 1}] // Flatten (* Jean-François Alcover, Feb 14 2019 *)

Crossrefs

Sequence in context: A206490 A343315 A321302 * A367072 A251548 A316785

Adjacent sequences: A294732 A294733 A294734 * A294736 A294737 A294738

Keyword

nonn,tabl

Author

Eric M. Schmidt, Nov 07 2017

Status

approved