A075498 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-18 + 3\,x + 2\,x^{2}\right)\,P_{n-2}(x) + \left(-9\,x + 6\,x^{2}\right)\,P_{n-2}'(x) + \left(9 - x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	1							5
2	18	11	1						30
3	90	99	21	1					211
4	486	855	315	34	1				1691
5	2754	7371	4185	765	50	1			15126
6	16038	64071	52731	14490	1575	69	1		148975
7	94770	562059	648081	251181	40320	2898	91	1	1599401

All 15 rows

Technical details

Node id: n-46955655870b4332 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1c283a95c09a98e0

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,        1;
  2:      18,       11,        1;
  3:      90,       99,       21,        1;
  4:     486,      855,      315,       34,        1;
  5:    2754,     7371,     4185,      765,       50,       1;
  6:   16038,    64071,    52731,    14490,     1575,      69,      1;
  7:   94770,   562059,   648081,   251181,    40320,    2898,     91,    1;
  8:  564246,  4968135,  7869555,  4140234,   920241,   96390,   4914,  116,   1;
  9: 3372354, 44162091, 94992345, 66196845, 19578510, 2801547, 206010, 7830, 144, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(18-3*x-2*x^2)*P(n-2,x) - (9*x-6*x^2)*d/dx P(n-2,x) + (9-x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-(18 - 3*x - 2*x^2)*P[n - 2, x] - (9*x - 6*x^2)*D[P[n - 2, x], x] + (9 - x)*P[n - 1, x] + 3*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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(18 - 3*x - 2*x^2)*v[i - 2] - (9*x - 6*x^2)*deriv(v[i - 2], x) + (9 - x)*v[i - 1] + 3*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
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 Stirling2 triangle with scaled diagonals (powers of 3).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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.