A099599 Triangle T read by rows: coefficients of polynomials generating array A099597.

1, 1, 1, 1, 0, 2, 1, 9, -12, 6, 1, -104, 204, -120, 24, 1, 2265, -4840, 3540, -1080, 120, 1, -71064, 164910, -138840, 54360, -10080, 720, 1, 3079825, -7626948, 7134330, -3300360, 808920, -100800, 5040, 1, -176449776, 460982648, -468313104, 244938960, -72266880, 12156480, -1088640, 40320

Offset: 0

Author

Ralf Stephan, Oct 28 2004

Keywords

• sign
• tabl
• changed

Comments

Row sums are k (A000027), left edge columns are factorials (A000142). - Peter Bala, Aug 19 2013

Examples

			Row polynomials:
  1,
  x + 1,
  2*x^2 + 1,
  6*x^3 - 12*x^2 + 9*x + 1,
  24*x^4 - 120*x^3 + 204*x^2 - 104*x + 1,
  120*x^5 - 1080*x^4 + 3540*x^3 - 4840*x^2 + 2265*x + 1,
  ...
		

Crossrefs

Cf. A000027, A000142, A099597, A132393.

Programs

Maple

# Define row polynomials R(n, x) recursively:
R := proc(n, x) option remember; if n = 0 then 1 elif n = 1 then 1+x
else (n*x+1)*procname(n-1,x-1) - (n-1)*(x-1)*procname(n-2, x-2) fi end:
Trow := n -> PolynomialTools:-CoefficientList(R(n,x), x);
seq(Trow(n), n = 0..10); # Peter Bala, Aug 19 2013

Mathematica

R[n_, x_] := R[n, x] = (n x + 1) R[n-1, x-1] - (n-1) (x-1) R[n-2, x-2]; R[0, ] = 1; R[1, x] = 1 + x;
Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 13 2019 *)
R[n_,y_] := HypergeometricPFQ[{1,-n,-y},{},1] (* Natalia L. Skirrow, Jul 18 2025 *)

Python

from fractions import Fraction as frac
from math import factorial as fact
from sympy.functions.combinatorial.numbers import stirling
A099599=lambda n,k: sum(map(lambda i: frac(fact(n),fact(n-i))*(-1)**(i-k)*stirling(i,k,kind=1),range(k,n+1))) # Natalia L. Skirrow, Jul 18 2025

Formula

The row polynomials satisfy the second order recurrence equation R(n,x) = (n*x+1)*R(n-1,x-1) - (n-1)*(x-1)*R(n-2,x-2), with the initial conditions R(0,x) = 1 and R(1,x) = 1+x. - Peter Bala, Aug 19 2013

From Natalia L. Skirrow, Jul 18 2025: (Start)

T(n,k) = Sum_{i=k..n} (n!/(n-i)!) * A048994(i,k).

T(n,k) = n * ((n-1)*(T(n-2,k)-T(n-1,k)) + T(n-1,k-1)) for k>0 (without this criterion, defines a unique continuation to negative k).

O.g.f.: hypergeom([1,1,-y],[],x/(x-1)) / (1-x).

E.g.f.: e^x*hypergeom([1,-y],[],-x).

Row polynomials are:

R(n,y) = hypergeom([1,-n,-y],[],1).

R(n,y) = n * ((y-n+1)*R(n-1,y) + (n-1)*R(n-2,y)) + 1.

R(n,y) = 1 + n!*y!*I_{y-n}(2) - hypergeom([1],[n+1,y+1],1) where I is the Bessel function. (End)