A257618 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-11 + 8\,x + 4\,n\right)\,P_{n}(x) = \left(140 - 888\,x + 3884\,x^{2} - 4272\,x^{3} + 896\,x^{4} - 80\,n + 640\,n x - 4464\,n x^{2} + 3712\,n x^{3} - 512\,n x^{4} - 128\,n^{2} x + 1856\,n^{2} x^{2} - 768\,n^{2} x^{3} - 256\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(112\,x - 1472\,x^{2} + 3216\,x^{3} - 2368\,x^{4} + 512\,x^{5} - 64\,n x + 1216\,n x^{2} - 1920\,n x^{3} + 768\,n x^{4} - 256\,n^{2} x^{2} + 256\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-124 + 492\,x - 408\,x^{2} + 64\,x^{3} + 48\,n - 320\,n x + 160\,n x^{2} + 64\,n^{2} x\right)\,P_{n-1}(x) + \left(-88\,x + 152\,x^{2} - 64\,x^{3} + 32\,n x - 32\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	44	48	4						96
3	480	1416	496	8					2400
4	4944	38912	31776	4992	16				80640
5	49856	917792	1735936	633280	49984	32			3386880
6	499648	19511040	77207232	60590080	12003648	499968	64		170311680
7	4999168	389170304	2921908992	4549705600	1870369280	222079872	4999936	128	9963233280

All 15 rows

Technical details

Node id: n-bac49424b4de44e2 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-84d0fce31cd67972

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:         4,            2;
  2:        44,           48,             4;
  3:       480,         1416,           496,              8;
  4:      4944,        38912,         31776,           4992,             16;
  5:     49856,       917792,       1735936,         633280,          49984,            32;
  6:    499648,     19511040,      77207232,       60590080,       12003648,        499968,            64;
  7:   4999168,    389170304,    2921908992,     4549705600,     1870369280,     222079872,       4999936,         128;
  8:  49998080,   7464984576,   98885690368,   279230050304,   220506324480,   53971025920,    4057478144,    49999872,       256;
  9: 499995648, 139769504256, 3100391882752, 14755366885376, 20617671763968, 9497792883712, 1497695657984, 73634715648, 499999744, 512;

%F Formula

The row polynomials satisfy (4*n-11+8*x)*P(n,x) = -(80*n-140+(128*n^2-640*n+888)*x+(256*n^3-1856*n^2+4464*n-3884)*x^2+(768*n^2-3712*n+4272)*x^3+(512*n-896)*x^4)*P(n-2,x) - ((64*n-112)*x+(256*n^2-1216*n+1472)*x^2-(256*n^2-1920*n+3216)*x^3-(768*n-2368)*x^4-512*x^5)*d/dx P(n-2,x) + (48*n-124+(64*n^2-320*n+492)*x+(160*n-408)*x^2+64*x^3)*P(n-1,x) + ((32*n-88)*x-(32*n-152)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(80*n - 140 + (128*n^2 - 640*n + 888)*x + (256*n^3 - 1856*n^2 + 4464*n - 3884)*x^2 + (768*n^2 - 3712*n + 4272)*x^3 + (512*n - 896)*x^4)*P[n - 2, x] - ((64*n - 112)*x + (256*n^2 - 1216*n + 1472)*x^2 - (256*n^2 - 1920*n + 3216)*x^3 - (768*n - 2368)*x^4 - 512*x^5)*D[P[n - 2, x], x] + (48*n - 124 + (64*n^2 - 320*n + 492)*x + (160*n - 408)*x^2 + 64*x^3)*P[n - 1, x] + ((32*n - 88)*x - (32*n - 152)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(4*n - 11 + 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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(80*n - 140 + (128*n^2 - 640*n + 888)*x + (256*n^3 - 1856*n^2 + 4464*n - 3884)*x^2 + (768*n^2 - 3712*n + 4272)*x^3 + (512*n - 896)*x^4)*v[i - 2] - ((64*n - 112)*x + (256*n^2 - 1216*n + 1472)*x^2 - (256*n^2 - 1920*n + 3216)*x^3 - (768*n - 2368)*x^4 - 512*x^5)*deriv(v[i - 2], x) + (48*n - 124 + (64*n^2 - 320*n + 492)*x + (160*n - 408)*x^2 + 64*x^3)*v[i - 1] + ((32*n - 88)*x - (32*n - 152)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(4*n - 11 + 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142459 → D+1

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.