A372873 → D

nonnegative coefficients

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

Recurrence

\(\left(48\,x + 2\,n - 80\,n x - 2\,n^{2} + 32\,n^{2} x\right)\,P_{n}(x) = \left(-3\,n + 52\,n x - 352\,n x^{2} + n^{2} - 28\,n^{2} x + 192\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2\,n x + 40\,n x^{2} - 128\,n x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	1	8													9
3	0	12	24												36
4	0	2	72	64											138
5	0	0	30	320	160										510
6	0	0	3	240	1200	384									1827
7	0	0	0	56	1400	4032	896								6384
8	0	0	0	4	560	6720	12544	2048							21876
9	0	0	0	0	90	4032	28224	36864	4608						73818
10	0	0	0	0	5	1080	23520	107520	103680	10240					246045
11	0	0	0	0	0	132	9240	118272	380160	281600	22528				811932
12	0	0	0	0	0	6	1848	63360	532224	1267200	743424	49152			2657214
13	0	0	0	0	0	0	182	18304	370656	2196480	4026880	1916928	106496		8635926
14	0	0	0	0	0	0	7	2912	144144	1921920	8456448	12300288	4845568	229376	27900663

Technical details

Node id: n-a3ee74063a2dcfe0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f11a2e82c5b46256

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 0;
  1: 2;
  2: 1,  8;
  3: 0, 12, 24;
  4: 0,  2, 72,  64;
  5: 0,  0, 30, 320,  160;
  6: 0,  0,  3, 240, 1200,  384;
  7: 0,  0,  0,  56, 1400, 4032,   896;
  8: 0,  0,  0,   4,  560, 6720, 12544,  2048;
  9: 0,  0,  0,   0,   90, 4032, 28224, 36864, 4608;

%F Formula

The row polynomials satisfy -(2*n^2-2*n-(32*n^2-80*n+48)*x)*P(n,x) = (n^2-3*n-(28*n^2-52*n)*x+(192*n^2-352*n)*x^2)*P(n-1,x) - (2*n*x-40*n*x^2+128*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 3*n - (28*n^2 - 52*n)*x + (192*n^2 - 352*n)*x^2)*P[n - 1, x] - (2*n*x - 40*n*x^2 + 128*n*x^3)*D[P[n - 1, x], x])/(-2*n^2 + 2*n + (32*n^2 - 80*n + 48)*x)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 3*n - (28*n^2 - 52*n)*x + (192*n^2 - 352*n)*x^2)*v[i - 1] - (2*n*x - 40*n*x^2 + 128*n*x^3)*deriv(v[i - 1], x))/(-2*n^2 + 2*n + (32*n^2 - 80*n + 48)*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 Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.?
  - 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.