A142461 → D

nonnegative coefficients

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

Recurrence

\(\left(-11 + 6\,n\right)\,P_{n}(x) = \left(35 - 170\,x + 935\,x^{2} - 42\,n + 264\,n x - 1962\,n x^{2} - 72\,n^{2} x + 1188\,n^{2} x^{2} - 216\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 360\,x^{2} + 330\,x^{3} - 36\,n x + 612\,n x^{2} - 576\,n x^{3} - 216\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-82 + 242\,x + 48\,n - 264\,n x + 72\,n^{2} x\right)\,P_{n-1}(x) + \left(-66\,x + 66\,x^{2} + 36\,n x - 36\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	14	2													16
3	111	222	3												336
4	796	5772	2388	4											8960
5	5597	105284	157926	22388	5										291200
6	39210	1648542	6001188	3297084	196050	6									11182080
7	274507	23862066	175842897	234457196	59655165	1647042	7								495738880
8	1921592	330520376	4450601112	11722859800	7417668520	991561128	13451144	8							24928583680
9	13451193	4462021800	102905301996	477029674296	596287092870	205810603992	15617076300	107609544	9						1402232832000
10	94158406	59324500314	2243001144024	17002403422536	36969799757940	25503605133804	5233669336056	237298001256	847425654	10					87250042880000
11	659108903	781575928742	46977372509535	553658817820776	1932424950788430	2318909940946116	968902931186358	125272993358760	3517091679339	6591089030	11				5950452924416000
12	4613762388	10240898359812	957050091802380	16910658782809020	89664334932308040	171599335630012584	125530068905231256	33821317565618040	2871150275407140	51204491799060	50751386268	12			441415416938496000
13	32296336789	133749922837548	19120993944168018	492950142969066700	3815372233348601715	10975855100817560472	12805164284287153884	6104595573357762744	1109137821680400075	63736646480560060	735624575606514	387556041468	13		35386802591236096000
14	226074357602	1743464262059318	376740752184365916	13878927748352332964	152166861562929988790	630450526051147518306	1101244128448695234024	840600701401530024408	273900350813273979822	34697319370880832410	1381382758009341692	10460785572355908	2938966648826	14	3048709146321879040000

Technical details

Node id: n-743f2f9ce8f29f74 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8db21305723b142a

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:        1;
  2:       14,          2;
  3:      111,        222,            3;
  4:      796,       5772,         2388,            4;
  5:     5597,     105284,       157926,        22388,            5;
  6:    39210,    1648542,      6001188,      3297084,       196050,            6;
  7:   274507,   23862066,    175842897,    234457196,     59655165,      1647042,           7;
  8:  1921592,  330520376,   4450601112,  11722859800,   7417668520,    991561128,    13451144,         8;
  9: 13451193, 4462021800, 102905301996, 477029674296, 596287092870, 205810603992, 15617076300, 107609544, 9;

%F Formula

The row polynomials satisfy (6*n-11)*P(n,x) = -(42*n-35+(72*n^2-264*n+170)*x+(216*n^3-1188*n^2+1962*n-935)*x^2)*P(n-2,x) - ((36*n-30)*x+(216*n^2-612*n+360)*x^2-(216*n^2-576*n+330)*x^3)*d/dx P(n-2,x) + (48*n-82+(72*n^2-264*n+242)*x)*P(n-1,x) + ((36*n-66)*x-(36*n-66)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(42*n - 35 + (72*n^2 - 264*n + 170)*x + (216*n^3 - 1188*n^2 + 1962*n - 935)*x^2)*P[n - 2, x] - ((36*n - 30)*x + (216*n^2 - 612*n + 360)*x^2 - (216*n^2 - 576*n + 330)*x^3)*D[P[n - 2, x], x] + (48*n - 82 + (72*n^2 - 264*n + 242)*x)*P[n - 1, x] + ((36*n - 66)*x - (36*n - 66)*x^2)*D[P[n - 1, x], x])/(6*n - 11)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(42*n - 35 + (72*n^2 - 264*n + 170)*x + (216*n^3 - 1188*n^2 + 1962*n - 935)*x^2)*v[i - 2] - ((36*n - 30)*x + (216*n^2 - 612*n + 360)*x^2 - (216*n^2 - 576*n + 330)*x^3)*deriv(v[i - 2], x) + (48*n - 82 + (72*n^2 - 264*n + 242)*x)*v[i - 1] + ((36*n - 66)*x - (36*n - 66)*x^2)*deriv(v[i - 1], x))/(6*n - 11)); 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) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
  - 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) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) +…?
    
    ↑ 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.