A089503 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(n - 7\,n x + 6\,n x^{2} + 4\,n^{2} x - 5\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-n x^{2} + 2\,n x^{3} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 0\)

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	4							4
3	12	60						72
4	24	336	1008					1368
5	40	1080	8640	20160				29920
6	60	2640	39600	237600	475200			755100
7	84	5460	131040	1441440	7207200	12972960		21758184
8	112	10080	352800	6115200	55036800	242161920	403603200	707280112

All 15 rows

Technical details

Node id: n-eb5c2993685be52f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d0e0bf31a1431b92

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:   4;
   3:  12,    60;
   4:  24,   336,    1008;
   5:  40,  1080,    8640,    20160;
   6:  60,  2640,   39600,   237600,     475200;
   7:  84,  5460,  131040,  1441440,    7207200,    12972960;
   8: 112, 10080,  352800,  6115200,   55036800,   242161920,   403603200;
   9: 144, 17136,  822528, 20563200,  287884800,  2245501440,  8982005760,  14114580480;
  10: 180, 27360, 1723680, 58605120, 1172102400, 14065228800, 98456601600, 365695948800, 548543923200;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n+(4*n^2-7*n)*x+(n^3-5*n^2+6*n)*x^2)*P(n-1,x) - (n*x^2+(n^2-2*n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n + (4*n^2 - 7*n)*x + (n^3 - 5*n^2 + 6*n)*x^2)*P[n - 1, x] - (n*x^2 + (n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((n + (4*n^2 - 7*n)*x + (n^3 - 5*n^2 + 6*n)*x^2)*v[i - 1] - (n*x^2 + (n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of numbers used for basis change between certain falling factorials.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.