A257621 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,n\right)\,P_{n}(x) = \left(21 - 54\,x + 45\,x^{2} - 84\,n + 240\,n x - 236\,n x^{2} - 96\,n^{2} x + 240\,n^{2} x^{2} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,x - 32\,x^{2} + 20\,x^{3} - 48\,n x + 144\,n x^{2} - 96\,n x^{3} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-38 + 50\,x + 40\,n - 80\,n x + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 20\,x^{2} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3														3
2	42	18													60
3	393	786	81												1260
4	3156	17292	9468	324											30240
5	23631	284892	427338	94524	1215										831600
6	171006	4031790	12820140	8063580	855030	4374									25945920
7	1216725	52215966	307208115	409610820	130539915	7300350	15309								908107200
8	8584872	640078776	6409572552	15565211160	10682620920	1920236328	60094104	52488							35286451200
9	60323739	7573128600	122066778708	492299236968	615374046210	244133557416	26505950100	482589912	177147						1508495788800
10	423033810	87527076330	2183152160520	13748089535880	28307206125660	20622134303820	5094021707880	350108305320	3807304290	590490					70396470144000
11	2963775777	995794476810	37342453915125	351450657149880	1115063762726130	1338076515271356	615038650012290	99579877107000	4481075145645	29637757770	1948617				3562061389286400
12	20754756348	11208623961732	618390785620260	8420210335220220	39265179507258840	72256131824653224	54971251310162376	16840420670440440	1855172356860780	56043119808660	228302319828	6377292			194294257597440000
13	145310397927	125245810675764	9998818029835614	192140317221437700	1271483330832837945	3411580236935665896	3980176943091610212	2034373329332540712	432315713748234825	33329393432785380	688851958716702	1743724775124	20726199		11366214069450240000
14	1017260473254	1392525578021958	158812100100175572	4223931594251224836	38610957073700360610	145514966292242172882	246770970471679833144	194019955056322897176	69499722732660649098	10559828985628062090	582311033700643764	8355153468131748	13224386152302	66961566	709951217261045760000

Technical details

Node id: n-4b0b2cb99a241d3f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-19a74063fe1c0664

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:        3;
  2:       42,         18;
  3:      393,        786,           81;
  4:     3156,      17292,         9468,          324;
  5:    23631,     284892,       427338,        94524,         1215;
  6:   171006,    4031790,     12820140,      8063580,       855030,         4374;
  7:  1216725,   52215966,    307208115,    409610820,    130539915,      7300350,       15309;
  8:  8584872,  640078776,   6409572552,  15565211160,  10682620920,   1920236328,    60094104,     52488;
  9: 60323739, 7573128600, 122066778708, 492299236968, 615374046210, 244133557416, 26505950100, 482589912, 177147;

%F Formula

The row polynomials satisfy (4*n-5)*P(n,x) = -(84*n-21+(96*n^2-240*n+54)*x+(64*n^3-240*n^2+236*n-45)*x^2)*P(n-2,x) - ((48*n-12)*x+(64*n^2-144*n+32)*x^2-(64*n^2-96*n+20)*x^3)*d/dx P(n-2,x) + (40*n-38+(32*n^2-80*n+50)*x)*P(n-1,x) + ((16*n-20)*x-(16*n-20)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(84*n - 21 + (96*n^2 - 240*n + 54)*x + (64*n^3 - 240*n^2 + 236*n - 45)*x^2)*P[n - 2, x] - ((48*n - 12)*x + (64*n^2 - 144*n + 32)*x^2 - (64*n^2 - 96*n + 20)*x^3)*D[P[n - 2, x], x] + (40*n - 38 + (32*n^2 - 80*n + 50)*x)*P[n - 1, x] + ((16*n - 20)*x - (16*n - 20)*x^2)*D[P[n - 1, x], x])/(4*n - 5)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n - 21 + (96*n^2 - 240*n + 54)*x + (64*n^3 - 240*n^2 + 236*n - 45)*x^2)*v[i - 2] - ((48*n - 12)*x + (64*n^2 - 144*n + 32)*x^2 - (64*n^2 - 96*n + 20)*x^3)*deriv(v[i - 2], x) + (40*n - 38 + (32*n^2 - 80*n + 50)*x)*v[i - 1] + ((16*n - 20)*x - (16*n - 20)*x^2)*deriv(v[i - 1], x))/(4*n - 5)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (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.