A257621 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9 + 8\,x + 4\,n\right)\,P_{n}(x) = \left(105 - 498\,x + 913\,x^{2} - 716\,x^{3} + 160\,x^{4} - 84\,n + 384\,n x - 860\,n x^{2} + 736\,n x^{3} - 128\,n x^{4} - 96\,n^{2} x + 368\,n^{2} x^{2} - 192\,n^{2} x^{3} - 64\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(60\,x - 336\,x^{2} + 644\,x^{3} - 496\,x^{4} + 128\,x^{5} - 48\,n x + 272\,n x^{2} - 416\,n x^{3} + 192\,n x^{4} - 64\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-78 + 206\,x - 164\,x^{2} + 32\,x^{3} + 40\,n - 128\,n x + 80\,n x^{2} + 32\,n^{2} x\right)\,P_{n-1}(x) + \left(-36\,x + 68\,x^{2} - 32\,x^{3} + 16\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	6	3							9
2	51	60	9						120
3	420	1179	474	27					2100
4	3237	20448	18114	3480	81				45360
5	23874	308523	569784	236970	24846	243			1164240
6	171735	4202796	14836035	12336960	2870925	175380	729		34594560
7	1218912	53432691	333316098	512013525	232942620	33408333	1232034	2187	1167566400

All 15 rows

Technical details

Node id: n-375df8450d6653e0 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cacd19d65f57e294

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:        1;
  1:        6,          3;
  2:       51,         60,            9;
  3:      420,       1179,          474,           27;
  4:     3237,      20448,        18114,         3480,           81;
  5:    23874,     308523,       569784,       236970,        24846,          243;
  6:   171735,    4202796,     14836035,     12336960,      2870925,       175380,         729;
  7:  1218912,   53432691,    333316098,    512013525,    232942620,     33408333,     1232034,       2187;
  8:  8591433,  648663648,   6729611940,  17701735344,  14573923710,   4056760512,   380133492,    8637360,     6561;
  9: 60343422, 7633452339, 125853343008, 532988163204, 738448855452, 367208366658, 67194876336, 4269154212, 60500886, 19683;

%F Formula

The row polynomials satisfy (4*n-9+8*x)*P(n,x) = -(84*n-105+(96*n^2-384*n+498)*x+(64*n^3-368*n^2+860*n-913)*x^2+(192*n^2-736*n+716)*x^3+(128*n-160)*x^4)*P(n-2,x) - ((48*n-60)*x+(64*n^2-272*n+336)*x^2-(64*n^2-416*n+644)*x^3-(192*n-496)*x^4-128*x^5)*d/dx P(n-2,x) + (40*n-78+(32*n^2-128*n+206)*x+(80*n-164)*x^2+32*x^3)*P(n-1,x) + ((16*n-36)*x-(16*n-68)*x^2-32*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(84*n - 105 + (96*n^2 - 384*n + 498)*x + (64*n^3 - 368*n^2 + 860*n - 913)*x^2 + (192*n^2 - 736*n + 716)*x^3 + (128*n - 160)*x^4)*P[n - 2, x] - ((48*n - 60)*x + (64*n^2 - 272*n + 336)*x^2 - (64*n^2 - 416*n + 644)*x^3 - (192*n - 496)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (40*n - 78 + (32*n^2 - 128*n + 206)*x + (80*n - 164)*x^2 + 32*x^3)*P[n - 1, x] + ((16*n - 36)*x - (16*n - 68)*x^2 - 32*x^3)*D[P[n - 1, x], x])/(4*n - 9 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(84*n - 105 + (96*n^2 - 384*n + 498)*x + (64*n^3 - 368*n^2 + 860*n - 913)*x^2 + (192*n^2 - 736*n + 716)*x^3 + (128*n - 160)*x^4)*v[i - 2] - ((48*n - 60)*x + (64*n^2 - 272*n + 336)*x^2 - (64*n^2 - 416*n + 644)*x^3 - (192*n - 496)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (40*n - 78 + (32*n^2 - 128*n + 206)*x + (80*n - 164)*x^2 + 32*x^3)*v[i - 1] + ((16*n - 36)*x - (16*n - 68)*x^2 - 32*x^3)*deriv(v[i - 1], x))/(4*n - 9 + 8*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
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) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ 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.