A163626 → D

Proved for all n all roots real interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	-1														-1
3	-3	4													1
4	-7	24	-18												-1
5	-15	100	-180	96											1
6	-31	360	-1170	1440	-600										-1
7	-63	1204	-6300	13440	-12600	4320									1
8	-127	3864	-30618	100800	-159600	120960	-35280								-1
9	-255	12100	-139860	667296	-1587600	1995840	-1270080	322560							1
10	-511	37320	-613890	4082400	-13696200	25401600	-26460000	14515200	-3265920						-1
11	-1023	114004	-2623500	23686080	-107692200	276423840	-419126400	372556800	-179625600	36288000					1
12	-2047	346104	-11007018	132422400	-794191200	2710350720	-5610472560	7185024000	-5568393600	2395008000	-439084800				-1
13	-4095	1046500	-45585540	720816096	-5592787200	24690631680	-67018311360	115960965120	-128432304000	88216128000	-34248614400	5748019200			1
14	-8191	3155880	-187051410	3847203360	-38061823800	213102489600	-737786649600	1656387532800	-2458427811840	2397403008000	-1478398521600	523069747200	-80951270400		-1
15	-16383	9500404	-762407100	20233624320	-252415963800	1765762558560	-7642630195200	21652965734400	-41355201888000	53687297664000	-46749358656000	26153487360000	-8499883392000	1220496076800	1

Technical details

Node id: n-8a7a8aaa440e4f07 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a6bffaf3050004e1

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 -2)
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:    0;
   2:   -1;
   3:   -3,     4;
   4:   -7,    24,     -18;
   5:  -15,   100,    -180,      96;
   6:  -31,   360,   -1170,    1440,      -600;
   7:  -63,  1204,   -6300,   13440,    -12600,     4320;
   8: -127,  3864,  -30618,  100800,   -159600,   120960,    -35280;
   9: -255, 12100, -139860,  667296,  -1587600,  1995840,  -1270080,   322560;
  10: -511, 37320, -613890, 4082400, -13696200, 25401600, -26460000, 14515200, -3265920;

%F Formula

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

%t Mathematica

P[1, x_] := 0;
P[2, x_] := -1;
P[n_, x_] := P[n, x] = Expand[-(2 - 4*x + 2*x^2)*P[n - 2, x] - (x - 2*x^2 + x^3)*D[P[n - 2, x], x] + (3 - 4*x)*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] = 0; v[2] = -1; for(n = 3, N, my(i = n); v[i] = -(2 - 4*x + 2*x^2)*v[i - 2] - (x - 2*x^2 + x^3)*deriv(v[i - 2], x) + (3 - 4*x)*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.