A156366 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-4 - 2\,n - 12\,n x - 18\,n x^{2} - 6\,n^{2} x - 27\,n^{2} x^{2} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 18\,x^{3} - n x - 6\,n x^{2} + 27\,n x^{3} - 3\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 6\,x + 3\,n + 12\,n x + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(x - 3\,x^{2} + n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 12 + 18\,x\)

\(P_{1}(x) = 33 + 198\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	12	18															30
1	33	198	81														312
2	78	1188	2106	324													3696
3	171	5436	24462	18468	1215												49752
4	360	21438	195696	385884	145800	4374											753552
5	741	77274	1265139	5060556	5215995	1080378	15309										12715392
6	1506	262944	7146954	50605560	107204310	63895392	7685118	52488									236854272
7	3039	861120	36870552	424554696	1592080110	1991009808	732382560	53170344	177147								4831109376
8	6108	2747466	178482528	3155148936	19104961320	42594510636	33733197792	8011610856	360671292	590490							107141927424
9	12249	8608878	825202485	21487185576	197452427490	710828738964	1015269518466	534731210280	84724272837	2410970670	1948617						2567740096512
10	24534	26635068	3688209450	137143217340	1829330132220	9951602345496	23049559665972	22217201209080	8066114067150	873763405740	15935789826	6377292					66144371079168
11	49107	81671652	16071273666	833244464700	15604564378545	122345924226408	428210734792428	674117181153144	455576411074725	117159585025140	8841487194738	104389892748	20726199				1822809695923200
12	98256	248845302	68695363872	4873358364804	124874053697520	1362009355449714	6851371756678848	16344112265396568	18206637029098416	8881695619855290	1652604368668704	88164397426788	678824469648	66961566			53517090040375296
13	196557	754789410	289338905655	27664169289876	950270354733285	14029531421258718	97714268750096775	335020350000331800	568196022560978079	461831392400376510	166379230151636733	22780230720029460	869106417486255	4387254842754	215233605		1667802744510185472
14	393162	2282058360	1204907355450	153366469543440	6946509746444130	135837425996249112	1272747749744829330	6030366300005972400	14727509675618739390	18338052509493630120	11140812331347095694	3018712220023529520	308310489205190550	8489455638277320	28207224869670	688747536	54987967448392925184

Technical details

Node id: n-a608f49c6fcfaa74 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-447106697bbaa40b

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 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:    12,      18;
  1:    33,     198,        81;
  2:    78,    1188,      2106,         324;
  3:   171,    5436,     24462,       18468,         1215;
  4:   360,   21438,    195696,      385884,       145800,         4374;
  5:   741,   77274,   1265139,     5060556,      5215995,      1080378,         15309;
  6:  1506,  262944,   7146954,    50605560,    107204310,     63895392,       7685118,        52488;
  7:  3039,  861120,  36870552,   424554696,   1592080110,   1991009808,     732382560,     53170344,      177147;
  8:  6108, 2747466, 178482528,  3155148936,  19104961320,  42594510636,   33733197792,   8011610856,   360671292,     590490;
  9: 12249, 8608878, 825202485, 21487185576, 197452427490, 710828738964, 1015269518466, 534731210280, 84724272837, 2410970670, 1948617;

%F Formula

The row polynomials satisfy (n+1)*P(n,x) = -(2*n+4+(6*n^2+12*n)*x+(9*n^3+27*n^2+18*n)*x^2)*P(n-2,x) - ((n+2)*x+(3*n^2+6*n)*x^2-(9*n^2+27*n+18)*x^3)*d/dx P(n-2,x) + (3*n+4+(6*n^2+12*n+6)*x)*P(n-1,x) + ((n+1)*x-(3*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 18*x, P(1,x) = 33 + 198*x + 81*x^2.

%t Mathematica

P[0, x_] := 12 + 18*x;
P[1, x_] := 33 + 198*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (6*n^2 + 12*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (3*n^2 + 6*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*D[P[n - 2, x], x] + (3*n + 4 + (6*n^2 + 12*n + 6)*x)*P[n - 1, x] + ((n + 1)*x - (3*n + 3)*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] = 12 + 18*x; v[2] = 33 + 198*x + 81*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (6*n^2 + 12*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*v[i - 2] - ((n + 2)*x + (3*n^2 + 6*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*deriv(v[i - 2], x) + (3*n + 4 + (6*n^2 + 12*n + 6)*x)*v[i - 1] + ((n + 1)*x - (3*n + 3)*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
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 of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.