A382232 - OEIS

%I #23 Nov 10 2025 21:50:42

%S 1,1,1,1,1,3,3,1,1,9,26,26,9,1,1,23,165,387,387,165,23,1,1,53,860,

%T 4292,9194,9194,4292,860,53,1,1,115,3967,38885,160778,314654,314654,

%U 160778,38885,3967,115,1,1,241,17022,307454,2291375,8041695,14743812,14743812,8041695,2291375,307454,17022,241,1

%N Irregular triangle read by rows: T(n,k) = [x^k] (1+x) * A_n(x)^2, where A_n(x) is the n-th Eulerian polynomial.

%H Ryuichi Sakamoto, <a href="https://arxiv.org/abs/1904.10667">The h*-polynomial of the cut polytope of K_{2,m} in the lattice spanned by its vertices</a>, arXiv:1904.10667 [math.CO], 2019.

%H Ryuichi Sakamoto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Sakamoto/saka6.html">The h*-polynomial of the cut polytope of K_{2,m} in the lattice spanned by its vertices</a>, Journal of Integer Sequences, Vol. 23, 2020, #20.7.5.

%H OEIS Wiki, <a href="https://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>.

%F T(n,k) = T(n,2*n-1-k) for n > 0.

%e Irregular triangle begins:

%e   1,  1;

%e   1,  1;

%e   1,  3,   3,    1;

%e   1,  9,  26,   26,    9,    1;

%e   1, 23, 165,  387,  387,  165,   23,   1;

%e   1, 53, 860, 4292, 9194, 9194, 4292, 860, 53, 1;

%e   ...

%o (PARI) a(n) = sum(k=0, n, k!*stirling(n, k, 2)*(x-1)^(n-k));

%o T(n, k) = polcoef((1+x)*a(n)^2, k);

%o for(n=0, 7, for(k=0, 2*(n+0^n)-1, print1(T(n, k), ", ")));

%Y Row sums give A048617.

%Y Cf. A125300, A165889, A173018.

%K nonn,tabf

%O 0,6

%A _Seiichi Manyama_, Mar 19 2025