A262226 Eulerian numbers of type D, the primary type.

1, 1, 0, 1, 2, 1, 1, 10, 13, 0, 1, 36, 118, 36, 1, 1, 116, 846, 836, 121, 0, 1, 358, 5279, 11764, 5279, 358, 1, 1, 1086, 30339, 129844, 129879, 30318, 1093, 0, 1, 3272, 165820, 1242616, 2337542, 1242616, 165820, 3272, 1, 1, 9832, 878188, 10853944, 34706710, 34706584, 10854028, 878152, 9841, 0

Offset

0,5

Comments

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

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Eli Bagno, David Garber, Mordechai Novick, The Worpitzky identity for the groups of signed and even-signed permutations, arXiv:2004.03681 [math.CO], 2020.

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:
  1;
  1,   0;
  1,   2,    1;
  1,  10,   13,     0;
  1,  36,  118,    36,    1;
  1, 116,  846,   836,  121,   0;
  1, 358, 5279, 11764, 5279, 358, 1;
  ...

Mathematica

T[n_, k_] := (Sum[(-1)^(k-i+1)(2i-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, Sep 24 2018 *)

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, A262227.

Sequence in context: A348453 A345748 A153731 * A298158 A154989 A064307

Adjacent sequences: A262223 A262224 A262225 * A262227 A262228 A262229

Keyword

nonn,tabl

Author

Michel Marcus, Sep 15 2015

Status

approved