A152842 → reverse → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 9\,x + 2\,x^{2} - 13\,n - 13\,n x - 3\,n x^{2} + 4\,n^{2} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(54 + 54\,x + 18\,x^{2} + 2\,x^{3} - 141\,n - 178\,n x - 76\,n x^{2} - 11\,n x^{3} + 88\,n^{2} + 122\,n^{2} x + 57\,n^{2} x^{2} + 9\,n^{2} x^{3} - 16\,n^{3} - 24\,n^{3} x - 12\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(27 + 54\,x + 36\,x^{2} + 10\,x^{3} + x^{4} - 60\,n - 134\,n x - 104\,n x^{2} - 34\,n x^{3} - 4\,n x^{4} + 24\,n^{2} + 56\,n^{2} x + 46\,n^{2} x^{2} + 16\,n^{2} x^{3} + 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

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

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	4	2													6
3	15	14	3												32
4	24	44	24	4											96
5	81	180	138	44	5										448
6	108	342	408	228	60	6									1152
7	351	1242	1737	1228	465	90	7								5120
8	432	1944	3600	3544	2000	648	112	8							12288
9	1377	6696	13716	15432	10430	4344	1092	152	9						53248
10	1620	9450	23760	33720	29720	16860	6160	1400	180	10					122880
11	5103	31590	85455	132840	131310	86244	38150	11240	2115	230	11				524288
12	5832	41796	132840	246780	297360	243816	138768	54840	14760	2580	264	12			1179648
13	18225	137052	461214	917460	1200555	1088280	700756	323112	105975	24140	3630	324	13		4980736
14	20412	173502	666792	1532412	2347380	2528946	1970416	1123976	469476	141890	30184	4284	364	14	11010048

Technical details

Node id: n-ba04e3e0da3e4c62 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-3ec4d21aec155d88

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:    0;
  1:    1;
  2:    4,    2;
  3:   15,   14,     3;
  4:   24,   44,    24,     4;
  5:   81,  180,   138,    44,     5;
  6:  108,  342,   408,   228,    60,    6;
  7:  351, 1242,  1737,  1228,   465,   90,    7;
  8:  432, 1944,  3600,  3544,  2000,  648,  112,   8;
  9: 1377, 6696, 13716, 15432, 10430, 4344, 1092, 152, 9;

%F Formula

The row polynomials satisfy (4*n^2-13*n+9+(4*n^2-13*n+9)*x+(n^2-3*n+2)*x^2)*P(n,x) = -(16*n^3-88*n^2+141*n-54+(24*n^3-122*n^2+178*n-54)*x+(12*n^3-57*n^2+76*n-18)*x^2+(2*n^3-9*n^2+11*n-2)*x^3)*P(n-1,x) + (24*n^2-60*n+27+(56*n^2-134*n+54)*x+(46*n^2-104*n+36)*x^2+(16*n^2-34*n+10)*x^3+(2*n^2-4*n+1)*x^4)*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[(-(16*n^3 - 88*n^2 + 141*n - 54 + (24*n^3 - 122*n^2 + 178*n - 54)*x + (12*n^3 - 57*n^2 + 76*n - 18)*x^2 + (2*n^3 - 9*n^2 + 11*n - 2)*x^3)*P[n - 1, x] + (24*n^2 - 60*n + 27 + (56*n^2 - 134*n + 54)*x + (46*n^2 - 104*n + 36)*x^2 + (16*n^2 - 34*n + 10)*x^3 + (2*n^2 - 4*n + 1)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 13*n + 9 + (4*n^2 - 13*n + 9)*x + (n^2 - 3*n + 2)*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] = (-(16*n^3 - 88*n^2 + 141*n - 54 + (24*n^3 - 122*n^2 + 178*n - 54)*x + (12*n^3 - 57*n^2 + 76*n - 18)*x^2 + (2*n^3 - 9*n^2 + 11*n - 2)*x^3)*v[i - 1] + (24*n^2 - 60*n + 27 + (56*n^2 - 134*n + 54)*x + (46*n^2 - 104*n + 36)*x^2 + (16*n^2 - 34*n + 10)*x^3 + (2*n^2 - 4*n + 1)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 13*n + 9 + (4*n^2 - 13*n + 9)*x + (n^2 - 3*n + 2)*x^2)); 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 reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,..…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,..…?
    no recorded property covers this requirement
    
    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.