A176663 T(n, k) = [x^k] Sum_{j=0..n} j!*binomial(x, j), for 0 <= k <= n, triangle read by rows.
1, 1, 1, 1, 0, 1, 1, 2, -2, 1, 1, -4, 9, -5, 1, 1, 20, -41, 30, -9, 1, 1, -100, 233, -195, 76, -14, 1, 1, 620, -1531, 1429, -659, 161, -20, 1, 1, -4420, 11537, -11703, 6110, -1799, 302, -27, 1, 1, 35900, -98047, 106421, -61174, 20650, -4234, 519, -35, 1
Offset: 0
Examples
Triangle starts:
{1},
{1, 1},
{1, 0, 1},
{1, 2, -2, 1},
{1, -4, 9, -5, 1},
{1, 20, -41, 30, -9, 1},
{1, -100, 233, -195, 76, -14, 1},
{1, 620, -1531, 1429, -659, 161, -20, 1},
{1, -4420, 11537, -11703, 6110, -1799, 302, -27, 1},
{1, 35900, -98047, 106421, -61174, 20650, -4234, 519, -35, 1},
{1, -326980, 928529, -1066279, 662506, -248675, 59039, -8931, 835, -44, 1}
Crossrefs
Programs
-
Maple
with(PolynomialTools): T_row := n -> CoefficientList(expand(add(k!*binomial(x, k), k=0..n)), x): ListTools:-Flatten([seq(T_row(n), n=0..9)]); # Peter Luschny, Jul 02 2019
-
Mathematica
p[x_, n_] := Sum[k! Binomial[x, k], {k, 0, n}]; Table[CoefficientList[FunctionExpand[p[x, n]], x], {n, 0, 10}] // Flatten (* Alternative: *) Table[CoefficientList[FunctionExpand[Sum[FactorialPower[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten (* Peter Luschny, Jul 02 2019 *)
Formula
From Peter Luschny, Jul 02 2019: (Start)
Sum_{k=0..n} T(n, k)*x^k = Sum_{k=0..n} (x)_k, where (x)_k denotes the falling factorial.
Let T be the lower triangular matrix associated to the T(n, k) and S the lower triangular matrix associated to the Stirling set numbers S2(n, k). Then S*T = A186020 (seen as a matrix) and T*S = A000012 (seen as a matrix). (End)
T(n, k) = Sum_{i=0..n-k} Stirling1(i+k, k). - Igor Victorovich Statsenko, May 25 2024
Extensions
Edited by Peter Luschny, Jul 02 2019