A257623 → D

nonnegative coefficients

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

Recurrence

\(\left(-7 + 5\,n\right)\,P_{n}(x) = \left(48 - 144\,x + 168\,x^{2} - 120\,n + 420\,n x - 610\,n x^{2} - 150\,n^{2} x + 525\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 100\,x^{2} + 70\,x^{3} - 75\,n x + 300\,n x^{2} - 225\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-62 + 98\,x + 55\,n - 140\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-35\,x + 35\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	48	18													66
3	501	1002	81												1584
4	4494	26052	13482	324											44352
5	37815	500460	750690	151260	1215										1441440
6	309324	8245470	27024840	16490940	1546620	4374									53621568
7	2498649	124513254	770915835	1027887780	311283135	14991894	15309								2252105856
8	20072298	1783583136	19106041698	47282837880	31843402830	5350749408	140506086	52488							105527245824
9	160860507	24712075416	432196115796	1800797494632	2250996868290	864392231592	86492263956	1286884056	177147						5461034971392
10	1287828840	335090984010	9188118383040	60434971471560	126055824624240	90652457207340	21438942893760	1340363936040	11590459560	590490					309458648378880
11	10305760317	4479814360770	186999499363365	1855535675253240	6022505662680570	7227006795216684	3247187431693170	498665331635640	20159164623465	103057603170	1948617				19062652740139008
12	82456357062	59329997283612	3688537622516010	53398625160990780	256768103129584860	476982448484301144	359475344381418804	106797250321981560	11065612867548030	296649986418060	907019927682	6377292			1268532891434704896
13	659684337279	780854902106148	71110411989335334	1464234786543812580	10059682990031630505	27449046302691007368	32023887353139508596	16095492784050608808	3294528269723578305	237034706631117780	4294701961583814	7916212047348	20726199		90700101737581400064
14	5277583112196	10234233953877078	1347921792291270888	38702535871087386276	369525210913514408940	1424898170274435377202	2433815438838602653296	1899864227032580502936	665145379644325936092	96756339677718465690	4942379905067993256	61405403723262468	68608580458548	66961566	6935069317473531666432

Technical details

Node id: n-96f444dda9cea45a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2e8e7edac39f7fae

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:        48,          18;
  3:       501,        1002,           81;
  4:      4494,       26052,        13482,           324;
  5:     37815,      500460,       750690,        151260,          1215;
  6:    309324,     8245470,     27024840,      16490940,       1546620,         4374;
  7:   2498649,   124513254,    770915835,    1027887780,     311283135,     14991894,       15309;
  8:  20072298,  1783583136,  19106041698,   47282837880,   31843402830,   5350749408,   140506086,      52488;
  9: 160860507, 24712075416, 432196115796, 1800797494632, 2250996868290, 864392231592, 86492263956, 1286884056, 177147;

%F Formula

The row polynomials satisfy (5*n-7)*P(n,x) = -(120*n-48+(150*n^2-420*n+144)*x+(125*n^3-525*n^2+610*n-168)*x^2)*P(n-2,x) - ((75*n-30)*x+(125*n^2-300*n+100)*x^2-(125*n^2-225*n+70)*x^3)*d/dx P(n-2,x) + (55*n-62+(50*n^2-140*n+98)*x)*P(n-1,x) + ((25*n-35)*x-(25*n-35)*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[(-(120*n - 48 + (150*n^2 - 420*n + 144)*x + (125*n^3 - 525*n^2 + 610*n - 168)*x^2)*P[n - 2, x] - ((75*n - 30)*x + (125*n^2 - 300*n + 100)*x^2 - (125*n^2 - 225*n + 70)*x^3)*D[P[n - 2, x], x] + (55*n - 62 + (50*n^2 - 140*n + 98)*x)*P[n - 1, x] + ((25*n - 35)*x - (25*n - 35)*x^2)*D[P[n - 1, x], x])/(5*n - 7)]];
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] = (-(120*n - 48 + (150*n^2 - 420*n + 144)*x + (125*n^3 - 525*n^2 + 610*n - 168)*x^2)*v[i - 2] - ((75*n - 30)*x + (125*n^2 - 300*n + 100)*x^2 - (125*n^2 - 225*n + 70)*x^3)*deriv(v[i - 2], x) + (55*n - 62 + (50*n^2 - 140*n + 98)*x)*v[i - 1] + ((25*n - 35)*x - (25*n - 35)*x^2)*deriv(v[i - 1], x))/(5*n - 7)); 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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
  - 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(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.?
    
    ↑ 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.