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

%I A099599 #49 Aug 28 2025 20:00:00
%S A099599 1,1,1,1,0,2,1,9,-12,6,1,-104,204,-120,24,1,2265,-4840,3540,-1080,120,
%T A099599 1,-71064,164910,-138840,54360,-10080,720,1,3079825,-7626948,7134330,
%U A099599 -3300360,808920,-100800,5040,1,-176449776,460982648,-468313104,244938960,-72266880,12156480,-1088640,40320
%N A099599 Triangle T read by rows: coefficients of polynomials generating array A099597.
%C A099599 Row sums are k (A000027), left edge columns are factorials (A000142). - _Peter Bala_, Aug 19 2013
%F A099599 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
%F A099599 From _Natalia L. Skirrow_, Jul 18 2025: (Start)
%F A099599 T(n,k) = Sum_{i=k..n} (n!/(n-i)!) * A048994(i,k).
%F A099599 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).
%F A099599 O.g.f.: hypergeom([1,1,-y],[],x/(x-1)) / (1-x).
%F A099599 E.g.f.: e^x*hypergeom([1,-y],[],-x).
%F A099599 Row polynomials are:
%F A099599 R(n,y) = hypergeom([1,-n,-y],[],1).
%F A099599 R(n,y) = n * ((y-n+1)*R(n-1,y) + (n-1)*R(n-2,y)) + 1.
%F A099599 R(n,y) = 1 + n!*y!*I_{y-n}(2) - hypergeom([1],[n+1,y+1],1) where I is the Bessel function. (End)
%e A099599 Row polynomials:
%e A099599   1,
%e A099599   x + 1,
%e A099599   2*x^2 + 1,
%e A099599   6*x^3 - 12*x^2 + 9*x + 1,
%e A099599   24*x^4 - 120*x^3 + 204*x^2 - 104*x + 1,
%e A099599   120*x^5 - 1080*x^4 + 3540*x^3 - 4840*x^2 + 2265*x + 1,
%e A099599   ...
%p A099599 # Define row polynomials R(n, x) recursively:
%p A099599 R := proc(n, x) option remember; if n = 0 then 1 elif n = 1 then 1+x
%p A099599 else (n*x+1)*procname(n-1,x-1) - (n-1)*(x-1)*procname(n-2, x-2) fi end:
%p A099599 Trow := n -> PolynomialTools:-CoefficientList(R(n,x), x);
%p A099599 seq(Trow(n), n = 0..10); # _Peter Bala_, Aug 19 2013
%t A099599 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;
%t A099599 Table[CoefficientList[R[n, x], x], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Nov 13 2019 *)
%t A099599 R[n_,y_] := HypergeometricPFQ[{1,-n,-y},{},1] (* _Natalia L. Skirrow_, Jul 18 2025 *)
%o A099599 (Python)
%o A099599 from fractions import Fraction as frac
%o A099599 from math import factorial as fact
%o A099599 from sympy.functions.combinatorial.numbers import stirling
%o A099599 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
%Y A099599 Cf. A000027, A000142, A099597, A132393.
%K A099599 sign,tabl,changed
%O A099599 0,6
%A A099599 _Ralf Stephan_, Oct 28 2004