A257617 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 7\,n\right)\,P_{n}(x) = \left(90 - 380\,x + 1140\,x^{2} - 126\,n + 672\,n x - 2681\,n x^{2} - 196\,n^{2} x + 1764\,n^{2} x^{2} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(70\,x - 490\,x^{2} + 420\,x^{3} - 98\,n x + 931\,n x^{2} - 833\,n x^{3} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-118 + 288\,x + 77\,n - 336\,n x + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-84\,x + 84\,x^{2} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	36	8													44
3	388	776	24												1188
4	3676	24832	11028	64											39600
5	33564	566408	849612	134256	160										1584000
6	303260	11076368	39082152	22152736	1516300	384									74131200
7	2732156	199663128	1397326056	1863101408	499157820	16392936	896								3978374400
8	24595932	3435039776	43219802892	111786084480	72033004820	10305119328	172171524	2048							240975820800
9	221378236	57469421480	1220768091732	5485759477072	6857199346340	2441536183464	201142975180	1771025888	4608						16265867904000
10	1992437404	945190619056	32474076853056	236191179027104	507432751629160	354286768540656	75772845990464	3780762476224	17931936636	10240					1210903499520000
11	17932010364	15382066767416	829135351478160	9293972596820928	31765526656769640	38118631988123568	16264452044436624	2211027603941760	69219300453372	179320103640	22528				98567544860928000
12	161388255068	248695277771072	20569864593820740	342938978418938880	1767815239347810840	3354439614954873984	2474941335086935176	685877956837877760	61709593781462220	1243476388855360	1775270805748	49152			8709786691347456000
13	1452494647868	4004623788637896	499965975657152796	12070890950699297200	90272239791242069880	255785635706100478704	298416574990450558488	144435583665987311808	27159504639073418700	1666553252190509320	22025430837508428	17429935774416	106496		830332997908457472000
14	13072452592668	64323809697639632	11973765359068104984	410123462184065584416	4320832762020274483060	17537515860538869126624	30438490649025956965776	23383354480718492168832	7777498971636494069508	1025308655460163961040	43903806316583051608	385942858185837792	169941883704684	229376	84949452862942187520000

Technical details

Node id: n-3618a761aebd57c3 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0c5574eb81f1a737

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:         2;
  2:        36,           8;
  3:       388,         776,            24;
  4:      3676,       24832,         11028,            64;
  5:     33564,      566408,        849612,        134256,           160;
  6:    303260,    11076368,      39082152,      22152736,       1516300,           384;
  7:   2732156,   199663128,    1397326056,    1863101408,     499157820,      16392936,          896;
  8:  24595932,  3435039776,   43219802892,  111786084480,   72033004820,   10305119328,    172171524,       2048;
  9: 221378236, 57469421480, 1220768091732, 5485759477072, 6857199346340, 2441536183464, 201142975180, 1771025888, 4608;

%F Formula

The row polynomials satisfy (7*n-12)*P(n,x) = -(126*n-90+(196*n^2-672*n+380)*x+(343*n^3-1764*n^2+2681*n-1140)*x^2)*P(n-2,x) - ((98*n-70)*x+(343*n^2-931*n+490)*x^2-(343*n^2-833*n+420)*x^3)*d/dx P(n-2,x) + (77*n-118+(98*n^2-336*n+288)*x)*P(n-1,x) + ((49*n-84)*x-(49*n-84)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(126*n - 90 + (196*n^2 - 672*n + 380)*x + (343*n^3 - 1764*n^2 + 2681*n - 1140)*x^2)*P[n - 2, x] - ((98*n - 70)*x + (343*n^2 - 931*n + 490)*x^2 - (343*n^2 - 833*n + 420)*x^3)*D[P[n - 2, x], x] + (77*n - 118 + (98*n^2 - 336*n + 288)*x)*P[n - 1, x] + ((49*n - 84)*x - (49*n - 84)*x^2)*D[P[n - 1, x], x])/(7*n - 12)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(126*n - 90 + (196*n^2 - 672*n + 380)*x + (343*n^3 - 1764*n^2 + 2681*n - 1140)*x^2)*v[i - 2] - ((98*n - 70)*x + (343*n^2 - 931*n + 490)*x^2 - (343*n^2 - 833*n + 420)*x^3)*deriv(v[i - 2], x) + (77*n - 118 + (98*n^2 - 336*n + 288)*x)*v[i - 1] + ((49*n - 84)*x - (49*n - 84)*x^2)*deriv(v[i - 1], x))/(7*n - 12)); 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.?
    
    ↑ 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.