A381888 Triangle read by rows: T(n, k) = (n + 1) * Sum_{j=k..n} binomial(n, j) * Eulerian1(j, j - k).

1, 2, 2, 3, 9, 3, 4, 28, 28, 4, 5, 75, 165, 75, 5, 6, 186, 786, 786, 186, 6, 7, 441, 3311, 6181, 3311, 441, 7, 8, 1016, 12888, 40888, 40888, 12888, 1016, 8, 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9, 10, 5110, 168670, 1312750, 3445510, 3445510, 1312750, 168670, 5110, 10

Offset: 0

Author

Peter Luschny, Mar 11 2025

Keywords

• nonn
• tabl

Comments

Consider A381706, the number of permutations of k chosen numbers in [n] with i-1 descents, as a sequence of squares of size 1x1, 2x2, 3x3, ..., as displayed in the example section of A381706. Conjecture: T(n, k) is the sum of column k+1 of the (n+1)th square; in other words: T(n, k) = Sum_{j=0..n} b(n+1, j+1, k+1).

Examples

			Triangle starts:
  [0] 1;
  [1] 2,    2;
  [2] 3,    9,     3;
  [3] 4,   28,    28,      4;
  [4] 5,   75,   165,     75,      5;
  [5] 6,  186,   786,    786,    186,      6;
  [6] 7,  441,  3311,   6181,   3311,    441,     7;
  [7] 8, 1016, 12888,  40888,  40888,  12888,  1016,    8;
  [8] 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9;
		

Crossrefs

Cf. A046802, A173018 (Eulerian1), A122045 (Euler), A058877 (column 1), A007526 (row sums), A381706 (generalized Eulerian).

Programs

Maple

T := (n, k) -> (n + 1)*add(binomial(n, j)*combinat:-eulerian1(j, j - k), j = k .. n):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;
# Using the e.g.f.:
egf := ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1))))/(exp(x*(y - 1)) - y)^2:
ser := simplify(series(egf, x, 10)):
seq(seq(n!*coeff(coeff(ser, x, n), y, k), k = 0..n), n = 0..9);

SageMath

# Using function eulerian1 from A173018.
def T(n: int, k: int) -> int:
    return (n + 1) * sum(binomial(n, j) * eulerian1(j, j-k) for j in (k..n))
def Trow(n) -> list[int]: return [T(n, k) for k in (0..n)]
for n in (0..8): print(f"{n}: ", Trow(n))

Formula

T(n, k) = n! * [y^k] [x^n] ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1)))) / (exp(x*(y - 1)) - y)^2.

Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * (n + 1) * Euler(n).

T(n, k) = (n + 1) * A046802(n, k).