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
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
Programs
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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);
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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).
Comments