A290595 → D

nonnegative coefficients

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

Recurrence

\(\left(-7 + 3\,n\right)\,P_{n}(x) = \left(160 + 640\,x + 280\,x^{2} - 276\,n - 912\,n x - 414\,n x^{2} + 153\,n^{2} + 396\,n^{2} x + 189\,n^{2} x^{2} - 27\,n^{3} - 54\,n^{3} x - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-96\,x + 12\,x^{2} + 84\,x^{3} + 108\,n x - 9\,n x^{2} - 99\,n x^{3} - 27\,n^{2} x + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(46 + 98\,x - 63\,n - 84\,n x + 18\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-21\,x + 21\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	2														2
3	19	8													27
4	222	294	24												540
5	3194	8256	2661	64											14175
6	55024	227132	157614	19340	160										459270
7	1107336	6537576	7688028	2222736	125835	384									17681895
8	25526192	201088824	352553880	181662560	26505750	769218	896								788107320
9	663605680	6651534752	16056935292	12757532800	3478572950	285223920	4525633	2048							39897933075
10	19213911360	236723438592	745481842488	835985913552	362399742000	58181791500	2870269038	25961112	4608						2260882874250
11	613280884160	9047720302848	35749786803480	53318952341760	33347889568980	8764678074600	887309507112	27592437600	146284695	10240					141757356215475
12	21396862829440	370335942486656	1782852563950272	3389041746002400	2866976728717080	1109766172862592	189296502921624	12672332157408	256816450710	813317390	22528				9742596481718100
13	810173435729280	16182836920550400	92772273748031232	217701475866386496	237916809904705920	126515282478064704	32421670089130728	3763739311875648	172485913941234	2334865208160	4474755021	49152			728259087008427975
14	33089464522419200	752574128084716800	5044300764595855872	14254566902376812608	19446972080702268960	13524844489663442112	4832480122991091984	859494831184668672	70317992654280432	2264597139339580	20860046683278	24411342756	106496		58820926258373028750
15	1450094174476057600	37133209195600832000	286680879189644569728	956460978213017439488	1586311514649784249840	1391186509318390434048	657697947975092307696	165464357721329468928	21149144154243577296	1252357241042577320	28924408793888212	183906578558064	132236519779	229376	5104816100280230709375

Technical details

Node id: n-8c6e500b88f7095e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d1e20b03e5a3729d

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:           2;
   3:          19,            8;
   4:         222,          294,           24;
   5:        3194,         8256,         2661,           64;
   6:       55024,       227132,       157614,        19340,          160;
   7:     1107336,      6537576,      7688028,      2222736,       125835,         384;
   8:    25526192,    201088824,    352553880,    181662560,     26505750,      769218,        896;
   9:   663605680,   6651534752,  16056935292,  12757532800,   3478572950,   285223920,    4525633,     2048;
  10: 19213911360, 236723438592, 745481842488, 835985913552, 362399742000, 58181791500, 2870269038, 25961112, 4608;

%F Formula

The row polynomials satisfy (3*n-7)*P(n,x) = -(27*n^3-153*n^2+276*n-160+(54*n^3-396*n^2+912*n-640)*x+(27*n^3-189*n^2+414*n-280)*x^2)*P(n-2,x) - ((27*n^2-108*n+96)*x+(9*n-12)*x^2-(27*n^2-99*n+84)*x^3)*d/dx P(n-2,x) + (18*n^2-63*n+46+(18*n^2-84*n+98)*x)*P(n-1,x) + ((9*n-21)*x-(9*n-21)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 153*n^2 + 276*n - 160 + (54*n^3 - 396*n^2 + 912*n - 640)*x + (27*n^3 - 189*n^2 + 414*n - 280)*x^2)*P[n - 2, x] - ((27*n^2 - 108*n + 96)*x + (9*n - 12)*x^2 - (27*n^2 - 99*n + 84)*x^3)*D[P[n - 2, x], x] + (18*n^2 - 63*n + 46 + (18*n^2 - 84*n + 98)*x)*P[n - 1, x] + ((9*n - 21)*x - (9*n - 21)*x^2)*D[P[n - 1, x], x])/(3*n - 7)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(27*n^3 - 153*n^2 + 276*n - 160 + (54*n^3 - 396*n^2 + 912*n - 640)*x + (27*n^3 - 189*n^2 + 414*n - 280)*x^2)*v[i - 2] - ((27*n^2 - 108*n + 96)*x + (9*n - 12)*x^2 - (27*n^2 - 99*n + 84)*x^3)*deriv(v[i - 2], x) + (18*n^2 - 63*n + 46 + (18*n^2 - 84*n + 98)*x)*v[i - 1] + ((9*n - 21)*x - (9*n - 21)*x^2)*deriv(v[i - 1], x))/(3*n - 7)); 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) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - 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.