A257615 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(-35 - 120\,x + 73\,x^{2} - 30\,x^{3} - 8\,x^{4} - 70\,n + 20\,n x - 90\,n x^{2} + 8\,n x^{3} - 16\,n x^{4} - 40\,n^{2} x + 4\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-10\,x - 28\,x^{2} + 42\,x^{3} - 20\,x^{4} + 16\,x^{5} - 20\,n x + 20\,n x^{2} - 24\,n x^{3} + 24\,n x^{4} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 48\,x - 6\,x^{2} + 8\,x^{3} + 24\,n - 4\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 10\,x^{2} - 8\,x^{3} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 10 + 5\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	10	5							15
2	95	120	25						240
3	840	2145	1090	125					4200
4	7005	32120	32010	8880	625				80640
5	55910	424805	744040	397150	68410	3125			1693440
6	431995	5136960	14636955	13533400	4442145	510120	15625		38707200
7	3258340	58156625	254582970	378491275	213139280	46569495	3727090	78125	958003200

All 15 rows

Technical details

Node id: n-e3bc7bf65fd31062 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-fc62581a8b3dd254

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:        10,          5;
  2:        95,        120,          25;
  3:       840,       2145,        1090,          125;
  4:      7005,      32120,       32010,         8880,          625;
  5:     55910,     424805,      744040,       397150,        68410,         3125;
  6:    431995,    5136960,    14636955,     13533400,      4442145,       510120,       15625;
  7:   3258340,   58156625,   254582970,    378491275,    213139280,     46569495,     3727090,      78125;
  8:  24136505,  626660880,  4036469140,   9117622400,   8193174030,   3053748880,   467678660,   26870880,    390625;
  9: 176377410, 6503002725, 59623749760, 195864093740, 265566410340, 157179989470, 40969844720, 4559683500, 192002410, 1953125;

%F Formula

The row polynomials satisfy (2*n-1+4*x)*P(n,x) = -(70*n+35+(40*n^2-20*n+120)*x+(8*n^3-4*n^2+90*n-73)*x^2+(24*n^2-8*n+30)*x^3+(16*n+8)*x^4)*P(n-2,x) - ((20*n+10)*x+(8*n^2-20*n+28)*x^2-(8*n^2-24*n+42)*x^3-(24*n-20)*x^4-16*x^5)*d/dx P(n-2,x) + (24*n-2+(8*n^2-4*n+48)*x+(20*n-6)*x^2+8*x^3)*P(n-1,x) + ((4*n-2)*x-(4*n-10)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 5*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 10 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 35 + (40*n^2 - 20*n + 120)*x + (8*n^3 - 4*n^2 + 90*n - 73)*x^2 + (24*n^2 - 8*n + 30)*x^3 + (16*n + 8)*x^4)*P[n - 2, x] - ((20*n + 10)*x + (8*n^2 - 20*n + 28)*x^2 - (8*n^2 - 24*n + 42)*x^3 - (24*n - 20)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (24*n - 2 + (8*n^2 - 4*n + 48)*x + (20*n - 6)*x^2 + 8*x^3)*P[n - 1, x] + ((4*n - 2)*x - (4*n - 10)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n - 1 + 4*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] = 10 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n + 35 + (40*n^2 - 20*n + 120)*x + (8*n^3 - 4*n^2 + 90*n - 73)*x^2 + (24*n^2 - 8*n + 30)*x^3 + (16*n + 8)*x^4)*v[i - 2] - ((20*n + 10)*x + (8*n^2 - 20*n + 28)*x^2 - (8*n^2 - 24*n + 42)*x^3 - (24*n - 20)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (24*n - 2 + (8*n^2 - 4*n + 48)*x + (20*n - 6)*x^2 + 8*x^3)*v[i - 1] + ((4*n - 2)*x - (4*n - 10)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n - 1 + 4*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) = 2*x + 5.?
  - 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) = 2*x + 5.?
    
    ↑ 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.