A049403 → D

nonnegative coefficients

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

Recurrence

\(\left(-2\,x + n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-6\,x - 4\,x^{2} + 2\,n + 3\,n x - 2\,n x^{2} + n^{2} + 3\,n^{2} x - n^{3}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 2\,n x - 4\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	3	2															5
1	3	12	3														18
2	0	30	30	4													64
3	0	30	135	60	5												230
4	0	0	315	420	105	6											846
5	0	0	315	1680	1050	168	7										3220
6	0	0	0	3780	6300	2268	252	8									12608
7	0	0	0	3780	23625	18900	4410	360	9								51084
8	0	0	0	0	51975	103950	48510	7920	495	10							212860
9	0	0	0	0	51975	374220	363825	110880	13365	660	11						914936
10	0	0	0	0	0	810810	1891890	1081080	231660	21450	858	12					4037760
11	0	0	0	0	0	810810	6621615	7567560	2837835	450450	33033	1092	13				18322408
12	0	0	0	0	0	0	14189175	37837800	25540515	6756750	825825	49140	1365	14			85200584
13	0	0	0	0	0	0	14189175	129729600	170270100	75675600	14864850	1441440	70980	1680	15		406243440
14	0	0	0	0	0	0	0	275675400	827026200	643242600	202161960	30630600	2413320	99960	2040	16	1981252096

Technical details

Node id: n-cb146e943380d277 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-95f8f0bdb419f7a5

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 3,  2;
  1: 3, 12,   3;
  2: 0, 30,  30,    4;
  3: 0, 30, 135,   60,     5;
  4: 0,  0, 315,  420,   105,      6;
  5: 0,  0, 315, 1680,  1050,    168,      7;
  6: 0,  0,   0, 3780,  6300,   2268,    252,      8;
  7: 0,  0,   0, 3780, 23625,  18900,   4410,    360,     9;
  8: 0,  0,   0,    0, 51975, 103950,  48510,   7920,   495,  10;
  9: 0,  0,   0,    0, 51975, 374220, 363825, 110880, 13365, 660, 11;

%F Formula

The row polynomials satisfy (n^2+n-(2*n+2)*x)*P(n,x) = -(n^3-n^2-2*n-(3*n^2+3*n-6)*x+(2*n+4)*x^2)*P(n-1,x) + ((2*n^2+2*n)*x-(4*n+8)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 2*n - (3*n^2 + 3*n - 6)*x + (2*n + 4)*x^2)*P[n - 1, x] + ((2*n^2 + 2*n)*x - (4*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + n - (2*n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 2*n - (3*n^2 + 3*n - 6)*x + (2*n + 4)*x^2)*v[i - 1] + ((2*n^2 + 2*n)*x - (4*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + n - (2*n + 2)*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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.