A322944 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 4\,x - x^{2} + n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(6\,x - 3\,x^{2} - 3\,n x\right)\,P_{n-2}'(x) + \left(2\,x + 2\,n\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	6	2													8
4	38	30	3												71
5	272	376	84	4											736
6	2200	4680	1740	180	5										8805
7	19920	60560	33120	5560	330	6									119496
8	199920	827120	620760	149800	14210	546	7								1812363
9	2204160	11979520	11793600	3823680	514080	31248	840	8							30347136
10	26490240	184101120	230254080	96022080	17117520	1460592	61656	1224	9						555508521
11	344736000	2998598400	4650979200	2419491200	547965600	61437600	3615360	112080	1710	10					11026937160
12	4829932800	51665644800	97517481600	61849603200	17271469600	2439249120	187368720	8054640	191070	2310	11				235768997871
13	72488908800	939641472000	2125365580800	1614413785600	543641683200	93903237120	9028958400	504102720	16513200	309320	3036	12			5399004554208
14	1160301542400	17998165785600	48167504985600	43194165414400	17239984361600	3563370370560	416538362240	28970610240	1227643560	31648760	479908	3900	13		131770261208781
15	19731353241600	362283918796800	1134849863347200	1187271640755200	553952253254400	134715833091840	18736108110720	1576927031680	82925322480	2756593840	57369312	718536	4914	14	3413203637638536

Technical details

Node id: n-e1002bc37f852cd6 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a39af1f3b69a86b4

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 -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        0;
   2:        1;
   3:        6,         2;
   4:       38,        30,         3;
   5:      272,       376,        84,        4;
   6:     2200,      4680,      1740,      180,        5;
   7:    19920,     60560,     33120,     5560,      330,       6;
   8:   199920,    827120,    620760,   149800,    14210,     546,     7;
   9:  2204160,  11979520,  11793600,  3823680,   514080,   31248,   840,    8;
  10: 26490240, 184101120, 230254080, 96022080, 17117520, 1460592, 61656, 1224, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-n-2+(2*n-4)*x+x^2)*P(n-2,x) - ((3*n-6)*x+3*x^2)*d/dx P(n-2,x) + (2*n+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(n^2 - n - 2 + (2*n - 4)*x + x^2)*P[n - 2, x] - ((3*n - 6)*x + 3*x^2)*D[P[n - 2, x], x] + (2*n + 2*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(n^2 - n - 2 + (2*n - 4)*x + x^2)*v[i - 2] - ((3*n - 6)*x + 3*x^2)*deriv(v[i - 2], x) + (2*n + 2*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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
[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.