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Triangle read by rows: T(n,k) (3 <= k < n) gives number of solutions to certain 1-loop scattering equations refined by MHV degree.
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%I #9 Feb 14 2019 12:39:15

%S 2,6,6,14,44,14,30,210,210,30,62,832,1812,832,62,126,2982,12012,12012,

%T 2982,126,254,10068,68322,124952,68322,10068,254,510,32730,352350,

%U 1065930,1065930,352350,32730,510,1022,103784,1700456,7972568,13103540,7972568,1700456,103784,1022

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

%C See Appendix E of Farrow and Lipstein.

%H Joseph A. Farrow and Arthur E. Lipstein, <a href="https://arxiv.org/abs/1705.07087">From 4d Ambitwistor Strings to On Shell Diagrams and Back</a>, arXiv:1705.07087 [hep-th], 2017.

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

%e Triangle begins

%e 2

%e 6, 6

%e 14, 44, 14

%e 30, 210, 210, 30

%e 62, 832, 1812, 832, 62

%e 126, 2982, 12012, 12012, 2982, 126

%e 254, 10068, 68322, 124952, 68322, 10068, 254

%e 510, 32730, 352350, 1065930, 1065930, 352350, 32730, 510

%e 1022, 103784, 1700456, 7972568, 13103540, 7972568, 1700456, 103784, 1022

%t e[n_, k_] := Sum[(-1)^j (k - j)^n Binomial[n + 1, j], {j, 0, k}];

%t T[n_, k_] := e[n - 1, k - 1] - e[n - 2, k - 1] - e[n - 2, k - 2];

%t Table[T[n, k], {n, 4, 12}, {k, 3, n - 1}] // Flatten (* _Jean-François Alcover_, Feb 14 2019 *)

%K nonn,tabl

%O 4,1

%A _Eric M. Schmidt_, Nov 07 2017