A257625 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(81 - 432\,x + 1053\,x^{2} - 990\,x^{3} + 216\,x^{4} - 54\,n + 324\,n x - 1098\,n x^{2} + 936\,n x^{3} - 144\,n x^{4} - 72\,n^{2} x + 468\,n^{2} x^{2} - 216\,n^{2} x^{3} - 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(54\,x - 396\,x^{2} + 810\,x^{3} - 612\,x^{4} + 144\,x^{5} - 36\,n x + 324\,n x^{2} - 504\,n x^{3} + 216\,n x^{4} - 72\,n^{2} x^{2} + 72\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-54 + 168\,x - 138\,x^{2} + 24\,x^{3} + 24\,n - 108\,n x + 60\,n x^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x + 54\,x^{2} - 24\,x^{3} + 12\,n x - 12\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	6	3														9
2	63	72	9													144
3	648	1863	702	27												3240
4	6237	43416	37098	6480	81											93312
5	57834	875043	1634904	639090	58806	243										3265920
6	527067	15898032	59185323	47909880	10317537	530712	729									134369280
7	4767660	270079191	1836699462	2840071005	1231298496	161249697	4780782	2187								6348948480
8	42994233	4394400336	51093630036	138997041984	112202639694	29360212560	2476620036	43040160	6561							338610585600
9	387243342	69527304099	1316404078848	5892325476084	8197571194812	3929929943058	669268294416	37667829492	387400806	19683						20113468784640
10	3486193911	1080084154680	32123769250797	224740141595568	504419328593478	411490760415840	127252343440890	14834882159568	569674223667	3486725352	59049					1316517956812800
11	31379110992	16577768605383	753927818410494	7927394346557199	27241120431294696	35449973621934606	18502256855380500	3908123030874006	323010294713496	8586984174219	31380882462	177147				94131033912115200
12	282423159189	252432013365288	17199538207354494	263724424618615056	1332049746427263207	2623592236501682928	2169571992233656212	769574826073961280	115728707772945459	6951902424349608	129181444273998	282429005040	531441			7298775552570163200
13	2541845102130	3823760027200131	384281477068243896	8391938469297782598	60344608503637498050	172536208863358552605	214507348317728447328	120058999540213792788	30250781029999929654	3340689938247738285	148473656002440360	1941111213939846	2541864234006	1594323		609969099750506496000
14	22876725493395	57722426002266336	8454134084157285903	258068581872442129512	2575849454400610876959	10343390294754254197080	18532617581948214480363	15534518883991239058704	6161825850652027349721	1140557267614833577872	94707747577211352525	3154625398861504776	29147172010013493	22876787671992	4782969	54653231337645382041600

Technical details

Node id: n-c63b8c85c56b948d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-04cae45d42f04794

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:         1;
  1:         6,           3;
  2:        63,          72,             9;
  3:       648,        1863,           702,            27;
  4:      6237,       43416,         37098,          6480,            81;
  5:     57834,      875043,       1634904,        639090,         58806,           243;
  6:    527067,    15898032,      59185323,      47909880,      10317537,        530712,          729;
  7:   4767660,   270079191,    1836699462,    2840071005,    1231298496,     161249697,      4780782,        2187;
  8:  42994233,  4394400336,   51093630036,  138997041984,  112202639694,   29360212560,   2476620036,    43040160,      6561;
  9: 387243342, 69527304099, 1316404078848, 5892325476084, 8197571194812, 3929929943058, 669268294416, 37667829492, 387400806, 19683;

%F Formula

The row polynomials satisfy (2*n-5+4*x)*P(n,x) = -(54*n-81+(72*n^2-324*n+432)*x+(72*n^3-468*n^2+1098*n-1053)*x^2+(216*n^2-936*n+990)*x^3+(144*n-216)*x^4)*P(n-2,x) - ((36*n-54)*x+(72*n^2-324*n+396)*x^2-(72*n^2-504*n+810)*x^3-(216*n-612)*x^4-144*x^5)*d/dx P(n-2,x) + (24*n-54+(24*n^2-108*n+168)*x+(60*n-138)*x^2+24*x^3)*P(n-1,x) + ((12*n-30)*x-(12*n-54)*x^2-24*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(54*n - 81 + (72*n^2 - 324*n + 432)*x + (72*n^3 - 468*n^2 + 1098*n - 1053)*x^2 + (216*n^2 - 936*n + 990)*x^3 + (144*n - 216)*x^4)*P[n - 2, x] - ((36*n - 54)*x + (72*n^2 - 324*n + 396)*x^2 - (72*n^2 - 504*n + 810)*x^3 - (216*n - 612)*x^4 - 144*x^5)*D[P[n - 2, x], x] + (24*n - 54 + (24*n^2 - 108*n + 168)*x + (60*n - 138)*x^2 + 24*x^3)*P[n - 1, x] + ((12*n - 30)*x - (12*n - 54)*x^2 - 24*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(54*n - 81 + (72*n^2 - 324*n + 432)*x + (72*n^3 - 468*n^2 + 1098*n - 1053)*x^2 + (216*n^2 - 936*n + 990)*x^3 + (144*n - 216)*x^4)*v[i - 2] - ((36*n - 54)*x + (72*n^2 - 324*n + 396)*x^2 - (72*n^2 - 504*n + 810)*x^3 - (216*n - 612)*x^4 - 144*x^5)*deriv(v[i - 2], x) + (24*n - 54 + (24*n^2 - 108*n + 168)*x + (60*n - 138)*x^2 + 24*x^3)*v[i - 1] + ((12*n - 30)*x - (12*n - 54)*x^2 - 24*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 4*x)); 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)

Related ?

Same fingerprint ?

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.