A136011 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	2	1							3
3	4	3							7
4	8	9	1						18
5	16	27	6						49
6	32	81	28	1					142
7	64	243	120	10					437
8	128	729	496	69	1				1423
9	256	2187	2016	410	15				4884
10	512	6561	8128	2261	145	1			17608
11	1024	19683	32640	11970	1155	21			66493
12	2048	59049	130816	61909	8281	272	1		262376
13	4096	177147	523776	315850	55755	2814	28		1079466
14	8192	531441	2096128	1598421	360865	25599	469	1	4621116

Technical details

Node id: n-cecb1919331c7542 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-aea4041b1413e99b

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:   1;
  2:   2,    1;
  3:   4,    3;
  4:   8,    9,    1;
  5:  16,   27,    6;
  6:  32,   81,   28,   1;
  7:  64,  243,  120,  10;
  8: 128,  729,  496,  69,  1;
  9: 256, 2187, 2016, 410, 15;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[x^2*P[n - 3, x] + 2*x*P[n - 2, x] + x^2*D[P[n - 2, x], x] + (2 - x)*P[n - 1, x] + x*D[P[n - 1, x], 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] = 1; v[3] = 2 + x; for(n = 3, N, my(i = n + 1); v[i] = x^2*v[i - 3] + 2*x*v[i - 2] + x^2*deriv(v[i - 2], x) + (2 - x)*v[i - 1] + x*deriv(v[i - 1], 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 Irregular triangle read by rows, Stirling numbers of the second kind: columns shifted to allow (1, 1, 2, 2, 3, 3, ...) …?
  - 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.