A075501 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-72 + 24\,x + 5\,x^{2}\right)\,P_{n-2}(x) + \left(-36\,x + 30\,x^{2}\right)\,P_{n-2}'(x) + \left(18 - 4\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 54 + 20\,x + x^{2}\)

\(P_{1}(x) = 468 + 324\,x + 39\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	54	20	1								75
1	468	324	39	1							832
2	4536	5040	1080	64	1						10721
3	47952	79056	26460	2700	95	1					156264
4	536544	1270080	616896	95760	5670	132	1				2525083
5	6205248	20948544	14125104	3111696	274680	10584	175	1			44676032
6	73063296	353652480	322392960	96453504	11866176	671328	18144	224	1		858118113
7	868361472	6081889536	7385411520	2913900480	478593360	37204272	1459080	29160	279	1	17766849160

All 15 rows

Technical details

Node id: n-65e2e6cbaaf8da19 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-3928142656cba953

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:           54,            20,             1;
  1:          468,           324,            39,             1;
  2:         4536,          5040,          1080,            64,            1;
  3:        47952,         79056,         26460,          2700,           95,           1;
  4:       536544,       1270080,        616896,         95760,         5670,         132,          1;
  5:      6205248,      20948544,      14125104,       3111696,       274680,       10584,        175,         1;
  6:     73063296,     353652480,     322392960,      96453504,     11866176,      671328,      18144,       224,       1;
  7:    868361472,    6081889536,    7385411520,    2913900480,    478593360,    37204272,    1459080,     29160,     279,     1;
  8:  10369949184,  106050954240,  170272751616,   86846411520,  18489889440,  1893253824,  100889712,   2898720,   44550,   340,   1;
  9: 124137059328, 1867739710464, 3953243314944, 2571661737216, 694925256960, 91028095488, 6321324240, 244590192, 5363820, 65340, 407, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(72-24*x-5*x^2)*P(n-2,x) - (36*x-30*x^2)*d/dx P(n-2,x) + (18-4*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 54 + 20*x + x^2, P(1,x) = 468 + 324*x + 39*x^2 + x^3.

%t Mathematica

P[0, x_] := 54 + 20*x + x^2;
P[1, x_] := 468 + 324*x + 39*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(72 - 24*x - 5*x^2)*P[n - 2, x] - (36*x - 30*x^2)*D[P[n - 2, x], x] + (18 - 4*x)*P[n - 1, x] + 6*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] = 54 + 20*x + x^2; v[2] = 468 + 324*x + 39*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(72 - 24*x - 5*x^2)*v[i - 2] - (36*x - 30*x^2)*deriv(v[i - 2], x) + (18 - 4*x)*v[i - 1] + 6*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 6).?
  - 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.