A163626 → D+1

Proved for all n all roots real interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-2 + 5\,x - 4\,x^{2} + x^{3}\right)\,P_{n-2}(x) + \left(-x + 3\,x^{2} - 3\,x^{3} + x^{4}\right)\,P_{n-2}'(x) + \left(3 - 5\,x + x^{2}\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	-1														-1
3	-2	1	2													1
4	-6	17	-6	-6												-1
5	-14	85	-130	36	24											1
6	-30	329	-990	1050	-240	-120										-1
7	-62	1141	-5698	11340	-9240	1800	720									1
8	-126	3737	-28686	90594	-134400	89040	-15120	-5040								-1
9	-254	11845	-133810	620676	-1420776	1678320	-937440	141120	40320							1
10	-510	36809	-595230	3877770	-12675600	22662360	-22226400	10735200	-1451520	-362880						-1
11	-1022	112981	-2566498	22811580	-101770680	254885400	-373055760	312681600	-133056000	16329600	3628800					1
12	-2046	344057	-10833966	128753394	-761085600	2551512480	-5158747440	6383527920	-4670265600	1776297600	-199584000	-39916800				-1
13	-4094	1042405	-45062290	705620916	-5412583176	23572074240	-62903206080	106386920640	-113937183360	73945872000	-25427001600	2634508800	479001600			1
14	-8190	3147689	-185473470	3784852890	-37100022960	205490124840	-702269568000	1550989440000	-2251379370240	2124244362240	-1238658220800	388669881600	-37362124800	-6227020800		-1
15	-16382	9484021	-757656898	19979488620	-247357557720	1715279365800	-7348336435440	20561161420800	-38648581171200	49092275232000	-41380628889600	21903545664000	-6320426112000	566658892800	87178291200	1

Technical details

Node id: n-489570c2b2a8d92c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-ec0e5cdcf61dbc5b

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 -1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    1;
   2:    0,    -1;
   3:   -2,     1,       2;
   4:   -6,    17,      -6,      -6;
   5:  -14,    85,    -130,      36,        24;
   6:  -30,   329,    -990,    1050,      -240,     -120;
   7:  -62,  1141,   -5698,   11340,     -9240,     1800,       720;
   8: -126,  3737,  -28686,   90594,   -134400,    89040,    -15120,    -5040;
   9: -254, 11845, -133810,  620676,  -1420776,  1678320,   -937440,   141120,    40320;
  10: -510, 36809, -595230, 3877770, -12675600, 22662360, -22226400, 10735200, -1451520, -362880;

%F Formula

The row polynomials satisfy P(n,x) = -(2-5*x+4*x^2-x^3)*P(n-2,x) - (x-3*x^2+3*x^3-x^4)*d/dx P(n-2,x) + (3-5*x+x^2)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -x;
P[n_, x_] := P[n, x] = Expand[-(2 - 5*x + 4*x^2 - x^3)*P[n - 2, x] - (x - 3*x^2 + 3*x^3 - x^4)*D[P[n - 2, x], x] + (3 - 5*x + x^2)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -x; for(n = 3, N, my(i = n); v[i] = -(2 - 5*x + 4*x^2 - x^3)*v[i - 2] - (x - 3*x^2 + 3*x^3 - x^4)*deriv(v[i - 2], x) + (3 - 5*x + x^2)*v[i - 1] + (x - x^2)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? step › direct check

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.