A166343 → D

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 12 + 2\,x\)

\(P_{2}(x) = 27 + 54\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	12	2															14
2	27	54	3														84
3	58	324	174	4													560
4	121	1436	2154	484	5												4200
5	248	5518	17232	11036	1240	6											35280
6	503	19530	110313	147084	48825	3018	7										329280
7	1014	65632	617022	1470840	1028370	196896	7098	8									3386880
8	2037	213120	3157224	12290376	15362970	6314448	745920	16296	9								38102400
9	4084	676026	15186336	90919304	184355640	136378956	35434784	2704104	36756	10							465696000
10	8179	2109758	69871695	616584104	1901677750	2282013300	1079022182	186324520	9493911	81790	11						6147187200
11	16370	6509212	311133150	3921380860	17575907540	31948186200	24606270556	7842761720	933399450	32546060	180070	12					87178291200
12	32753	19920516	1351934598	23755346300	149570979915	392366015784	457760351748	239313567864	53449529175	4506448660	109562838	393036	13				1322204083200
13	65520	60613126	5766307680	138605105604	1194367708240	4361928693570	7324165627968	5815904924760	2149861874832	346512764010	21143128160	363678756	851760	14			21358681344000
14	131055	183673938	24247186677	785286450900	9072026451495	44865913353870	104393576306229	119306944349976	67298870030805	18144052902990	2159537739975	96988746708	1193880597	1834770	15		366148823040000
15	262126	554953464	100845899406	4346716823568	66211455472470	433812142636824	1358584266244982	2147524998299568	1746751199457834	723020237728040	145665202039434	13040150470704	436998897426	3884674248	3931890	16	6639498657792000

Technical details

Node id: n-674032b73934f4eb · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-4c4143f463ed4f9a

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 0)
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:   12,       2;
   2:   27,      54,        3;
   3:   58,     324,      174,         4;
   4:  121,    1436,     2154,       484,          5;
   5:  248,    5518,    17232,     11036,       1240,          6;
   6:  503,   19530,   110313,    147084,      48825,       3018,          7;
   7: 1014,   65632,   617022,   1470840,    1028370,     196896,       7098,         8;
   8: 2037,  213120,  3157224,  12290376,   15362970,    6314448,     745920,     16296,       9;
   9: 4084,  676026, 15186336,  90919304,  184355640,  136378956,   35434784,   2704104,   36756,    10;
  10: 8179, 2109758, 69871695, 616584104, 1901677750, 2282013300, 1079022182, 186324520, 9493911, 81790, 11;

%F Formula

The row polynomials satisfy n*P(n,x) = -(2*n+2+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) - ((n+1)*x+(n^2-1)*x^2-(n^2+n)*x^3)*d/dx P(n-2,x) + (3*n+1+2*n^2*x)*P(n-1,x) + (n*x-n*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 12 + 2*x, P(2,x) = 27 + 54*x + 3*x^2.

%t Mathematica

P[1, x_] := 12 + 2*x;
P[2, x_] := 27 + 54*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 2, x], x] + (3*n + 1 + 2*n^2*x)*P[n - 1, x] + (n*x - n*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 12 + 2*x; v[2] = 27 + 54*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*v[i - 2] - ((n + 1)*x + (n^2 - 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 2], x) + (3*n + 1 + 2*n^2*x)*v[i - 1] + (n*x - n*x^2)*deriv(v[i - 1], x))/(n)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)

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.