A201637 → reverse (shift +1) → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,n - 8\,n x - 2\,n x^{2} + 3\,n^{2} + 8\,n^{2} x + 3\,n^{2} x^{2} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,n x - n x^{2} - n x^{3} - n^{2} x + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2\,x - 3\,n - 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	2	2													4
3	6	16	3												25
4	24	116	66	4											210
5	120	888	984	208	5										2205
6	720	7416	13200	5808	570	6									27720
7	5040	67968	174420	128480	28050	1440	7								405405
8	40320	682272	2355912	2576080	979000	119700	3458	8							6756750
9	362880	7467840	33078888	49760256	28827500	6375000	470820	8032	9						126351225
10	3628800	88678080	486560736	955619616	776786920	268590000	37283120	1750784	18234	10					2618916300
11	39916800	1136712960	7522443648	18568655808	20013475440	9890305824	2200588040	201951232	6255342	40720	11				59580345825
12	479001600	15655438080	122364905472	368799160896	505902165120	336248392704	108842725488	16411862208	1033887384	21706260	89826	12			1475759335050
13	6227020800	230672171520	2093788310400	7532318647296	12749327367120	10909131853824	4814040324688	1074587361792	114028359984	5072592760	73696656	196272	13		39529267903125
14	87178291200	3621985113600	37652712053760	158758312223232	323641875699360	344966240787552	198099370462784	61078657195136	9762088611744	750160522280	24088127888	246095856	425594	14	1138442915610000

Technical details

Node id: n-ff30b51583c3e145 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e6b802c8b5974669

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 -1)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      0;
  1:      1;
  2:      2,       2;
  3:      6,      16,        3;
  4:     24,     116,       66,        4;
  5:    120,     888,      984,      208,        5;
  6:    720,    7416,    13200,     5808,      570,       6;
  7:   5040,   67968,   174420,   128480,    28050,    1440,      7;
  8:  40320,  682272,  2355912,  2576080,   979000,  119700,   3458,    8;
  9: 362880, 7467840, 33078888, 49760256, 28827500, 6375000, 470820, 8032, 9;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = -(n^3-3*n^2+2*n+(2*n^3-8*n^2+8*n)*x+(n^3-3*n^2+2*n)*x^2)*P(n-2,x) - ((n^2-2*n)*x+n*x^2-(n^2-n)*x^3)*d/dx P(n-2,x) + (2*n^2-3*n+(2*n^2-4*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2*n + (2*n^3 - 8*n^2 + 8*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 2, x] - ((n^2 - 2*n)*x + n*x^2 - (n^2 - n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 3*n + (2*n^2 - 4*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 2*n + (2*n^3 - 8*n^2 + 8*n)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 2] - ((n^2 - 2*n)*x + n*x^2 - (n^2 - n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 3*n + (2*n^2 - 4*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Row reversed version of triangle A201637 (second-order Eulerian triangle).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.