A262227 Eulerian numbers of type D, the complementary type.

0, 0, 1, 0, 4, 0, 0, 13, 10, 1, 0, 40, 112, 40, 0, 0, 121, 836, 846, 116, 1, 0, 364, 5264, 11784, 5264, 364, 0, 0, 1093, 30318, 129879, 129844, 30339, 1086, 1, 0, 3280, 165792, 1242672, 2337472, 1242672, 165792, 3280, 0, 0, 9841, 878152, 10854028, 34706584, 34706710, 10853944, 878188, 9832, 1

Offset

0,5

Comments

Named D~(n, k) (the complementary type D triangle) in the Borowiec link.

Links

Table of n, a(n) for n=0..54.

Anna Borowiec, Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.

Katarzyna Kril, Wojciech Mlotkowski, Permutations of Type B with Fixed Number of Descents and Minus Signs, Volume 26(1) of The Electronic Journal of Combinatorics, 2019.

Formula

T(n, k) = (A060187(n+1, k+1) - (-1)^k*binomial(n, k))/2.

Example

Triangle begins:
  0;
  0,   1;
  0,   4,    0;
  0,  13,   10,     1;
  0,  40,  112,    40,    0;
  0, 121,  836,   846,  116,   1;
  0, 364, 5264, 11784, 5264, 364, 0;
  ...

Mathematica

T[n_, k_] := (Sum[(-1)^(k - i + 1)*(2*i - 1)^n*Binomial[n + 1, k - i + 1], {i, 1, k + 1}] - (-1)^k*Binomial[n, k])/2; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2019 *)

Prog

(PARI) B(n, k) = if( n<k || k<1, 0, sum(i=1, k, (-1)^(k-i) * binomial(n, k-i) * (2*i-1)^(n-1)));
T(n, k) = (A060187(n+1, k+1) - (-1)^k*binomial(n, k))/2;

Crossrefs

Cf. A060187, A262226.

Sequence in context: A178517 A049207 A092219 * A336731 A069026 A133851

Adjacent sequences: A262224 A262225 A262226 * A262228 A262229 A262230

Keyword

nonn,tabl

Author

Michel Marcus, Sep 15 2015

Status

approved