cp's OEIS Frontend

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

%I A381888 #20 Mar 15 2025 09:30:06
%S A381888 1,2,2,3,9,3,4,28,28,4,5,75,165,75,5,6,186,786,786,186,6,7,441,3311,
%T A381888 6181,3311,441,7,8,1016,12888,40888,40888,12888,1016,8,9,2295,47529,
%U A381888 241191,404361,241191,47529,2295,9,10,5110,168670,1312750,3445510,3445510,1312750,168670,5110,10
%N A381888 Triangle read by rows: T(n, k) = (n + 1) * Sum_{j=k..n} binomial(n, j) * Eulerian1(j, j - k).
%C A381888 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).
%F A381888 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.
%F A381888 Sum_{k=0..n} (-1)^k * T(n, k) = (-1)^n * (n + 1) * Euler(n).
%F A381888 T(n, k) = (n + 1) * A046802(n, k).
%e A381888 Triangle starts:
%e A381888   [0] 1;
%e A381888   [1] 2,    2;
%e A381888   [2] 3,    9,     3;
%e A381888   [3] 4,   28,    28,      4;
%e A381888   [4] 5,   75,   165,     75,      5;
%e A381888   [5] 6,  186,   786,    786,    186,      6;
%e A381888   [6] 7,  441,  3311,   6181,   3311,    441,     7;
%e A381888   [7] 8, 1016, 12888,  40888,  40888,  12888,  1016,    8;
%e A381888   [8] 9, 2295, 47529, 241191, 404361, 241191, 47529, 2295, 9;
%p A381888 T := (n, k) -> (n + 1)*add(binomial(n, j)*combinat:-eulerian1(j, j - k), j = k .. n):
%p A381888 for n from 0 to 8 do seq(T(n, k), k=0..n) od;
%p A381888 # Using the e.g.f.:
%p A381888 egf := ((y - 1)*(y*exp(x*y)*(x*y + 1) - (x + 1)*exp(x*(2*y - 1))))/(exp(x*(y - 1)) - y)^2:
%p A381888 ser := simplify(series(egf, x, 10)):
%p A381888 seq(seq(n!*coeff(coeff(ser, x, n), y, k), k = 0..n), n = 0..9);
%o A381888 (SageMath)  # Using function eulerian1 from A173018.
%o A381888 def T(n: int, k: int) -> int:
%o A381888     return (n + 1) * sum(binomial(n, j) * eulerian1(j, j-k) for j in (k..n))
%o A381888 def Trow(n) -> list[int]: return [T(n, k) for k in (0..n)]
%o A381888 for n in (0..8): print(f"{n}: ", Trow(n))
%Y A381888 Cf. A046802, A173018 (Eulerian1), A122045 (Euler), A058877 (column 1), A007526 (row sums), A381706 (generalized Eulerian).
%K A381888 nonn,tabl
%O A381888 0,2
%A A381888 _Peter Luschny_, Mar 11 2025