A200536 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 3 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	3	4													7
2	6	26	36	16											84
3	9	66	189	264	180	48									756
4	12	124	540	1284	1800	1488	672	128							6048
5	15	200	1170	3940	8415	11820	10920	6400	2160	320					45360
6	18	294	2160	9420	27090	53934	75852	75360	51840	23520	6336	768			326592
7	21	406	3591	19236	69615	179718	340473	479248	501228	384720	210672	77952	17472	1792	2286144

All 15 rows

Technical details

Node id: n-a35b6e0a133d5096 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-da2813591e6631ab

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  3,   4;
  2:  6,  26,   36,    16;
  3:  9,  66,  189,   264,    180,      48;
  4: 12, 124,  540,  1284,   1800,    1488,     672,     128;
  5: 15, 200, 1170,  3940,   8415,   11820,   10920,    6400,     2160,      320;
  6: 18, 294, 2160,  9420,  27090,   53934,   75852,   75360,    51840,    23520,     6336,      768;
  7: 21, 406, 3591, 19236,  69615,  179718,  340473,  479248,   501228,   384720,   210672,    77952,    17472,    1792;
  8: 24, 536, 5544, 35224, 153720,  488040, 1164408, 2125832,  2994192,  3253600,  2705472,  1690752,   768768,  240128,   46080,   4096;
  9: 27, 684, 8100, 59544, 304290, 1147104, 3302964, 7419528, 13163067, 18548820, 20761488, 18353664, 12658464, 6668928, 2592000, 700416, 117504, 9216;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + 3*n*x + 2*n*x^2)*P[n - 1, x])/(n - 1)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 3*n*x + 2*n*x^2)*v[i - 1])/(n - 1)); 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, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Parent-row provers of polytool polytool ? 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.