A397048 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,x - x^{2} + n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3\,x - 2\,x^{2} + n - 3\,n x - n^{2}\right)\,P_{n-2}'(x) + \left(x - x^{2} - n x\right)\,P_{n-2}''(x) + \left(1 + 2\,x + 2\,n\right)\,P_{n-1}(x) + \left(1 + 2\,x + n\right)\,P_{n-1}'(x) + \left(x\right)\,P_{n-1}''(x)\)

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

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

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	5	2													7
3	37	24	3												64
4	369	302	66	4											741
5	4651	4380	1275	140	5										10451
6	70993	73502	25650	3860	255	6									174266
7	1273609	1412208	559503	103320	9520	420	7								3358587
8	26269505	30683774	13360914	2856084	327810	20412	644	8							73519151
9	612579511	745368180	349258395	83414100	11227335	879984	39522	936	9						1802767972
10	15937424601	20042515022	9959336490	2595862420	394813125	36626562	2086560	70800	1305	10					48968736895
11	457696700077	591455323272	308398714503	86289288800	14466028610	1529015796	103815789	4494600	119295	1760	11				1459943502513
12	14381984674225	19013842578590	10321377841962	3062809893044	556080583080	65267718648	5080844076	263582616	8967915	191290	2310	12			47406716877768
13	490839666661891	661640628443340	371641817731275	115894324525980	22492460455465	2878205890560	250230028048	14957184528	612411228	16808220	294437	2964	13		1665652920437949
14	18080919199832609	24783748812704702	14335907848115490	4664654716006740	958116516191835	131851909066878	12563053783280	842477086880	39903293430	1322601280	29903874	437892	3731	14	62968645789028635

Technical details

Node id: n-f8120bd1c80841f1 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9b833f304e0df842

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:         5,         2;
  3:        37,        24,         3;
  4:       369,       302,        66,        4;
  5:      4651,      4380,      1275,      140,        5;
  6:     70993,     73502,     25650,     3860,      255,      6;
  7:   1273609,   1412208,    559503,   103320,     9520,    420,     7;
  8:  26269505,  30683774,  13360914,  2856084,   327810,  20412,   644,   8;
  9: 612579511, 745368180, 349258395, 83414100, 11227335, 879984, 39522, 936, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-n+(2*n-2)*x+x^2)*P(n-2,x) - (n^2-n+(3*n-3)*x+2*x^2)*d/dx P(n-2,x) - ((n-1)*x+x^2)*d^2/dx^2 P(n-2,x) + (2*n+1+2*x)*P(n-1,x) + (n+1+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 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[-(n^2 - n + (2*n - 2)*x + x^2)*P[n - 2, x] - (n^2 - n + (3*n - 3)*x + 2*x^2)*D[P[n - 2, x], x] - ((n - 1)*x + x^2)*D[P[n - 2, x], {x, 2}] + (2*n + 1 + 2*x)*P[n - 1, x] + (n + 1 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
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] = -(n^2 - n + (2*n - 2)*x + x^2)*v[i - 2] - (n^2 - n + (3*n - 3)*x + 2*x^2)*deriv(v[i - 2], x) - ((n - 1)*x + x^2)*deriv(deriv(v[i - 2], x), x) + (2*n + 1 + 2*x)*v[i - 1] + (n + 1 + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.