A079638 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 112\,n - 92\,n^{2} + 32\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(-36 + 8\,x + 78\,n - 12\,n x - 54\,n^{2} + 4\,n^{2} x + 12\,n^{3}\right)\,P_{n-3}(x) + \left(-13 + 6\,x - x^{2} + 26\,n - 6\,n x - 13\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 9 + 2\,x\)

\(P_{3}(x) = 83 + 36\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	9	2													11
3	83	36	3												122
4	870	550	90	4											1514
5	10474	8550	2055	180	5										21264
6	143892	141848	44415	5740	315	6									336216
7	2233356	2548392	972867	164640	13370	504	7								5933136
8	38769840	49741144	22196916	4605636	491400	27468	756	8							115833168
9	745785936	1053582120	533835600	130239900	17099565	1258740	51450	1080	9						2481854400
10	15766233120	24145629552	13599996300	3789963320	587334825	53337438	2875950	89760	1485	10					57945461760
11	363640940640	596550175056	367574490228	114549339960	20357599075	2188014444	145814361	6011280	148005	1980	11				1465012535040
12	9092247615360	15828641981568	10537107887592	3613325242384	721003847850	89116555242	7042749714	359354424	11698830	233090	2574	12			39888857168640
13	245065185624960	449377877894400	320006929637304	119229767568912	26284235707730	3658089765330	333142472663	20188159992	814350537	21471450	353353	3276	13		1163976253009920
14	7085176946023680	13602375561523968	10278850590524880	4119075755849600	990529708568400	152742426885048	15696194348835	1097831963800	52656028425	1721790070	37522485	518700	4095	14	36245599431552000

Technical details

Node id: n-47afaee86b7a6b82 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ca763dc598cfee6a

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 -1)
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:         0;
  1:         1;
  2:         9,          2;
  3:        83,         36,         3;
  4:       870,        550,        90,         4;
  5:     10474,       8550,      2055,       180,        5;
  6:    143892,     141848,     44415,      5740,      315,       6;
  7:   2233356,    2548392,    972867,    164640,    13370,     504,     7;
  8:  38769840,   49741144,  22196916,   4605636,   491400,   27468,   756,    8;
  9: 745785936, 1053582120, 533835600, 130239900, 17099565, 1258740, 51450, 1080, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^4-32*n^3+92*n^2-112*n+48)*P(n-4,x) + (12*n^3-54*n^2+78*n-36+(4*n^2-12*n+8)*x)*P(n-3,x) - (13*n^2-26*n+13+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 9 + 2*x, P(3,x) = 83 + 36*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 9 + 2*x;
P[3, x_] := 83 + 36*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*P[n - 4, x] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*P[n - 3, x] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 9 + 2*x; v[4] = 83 + 36*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(4*n^4 - 32*n^3 + 92*n^2 - 112*n + 48)*v[i - 4] + (12*n^3 - 54*n^2 + 78*n - 36 + (4*n^2 - 12*n + 8)*x)*v[i - 3] - (13*n^2 - 26*n + 13 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 2*x)*v[i - 1]); 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 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - 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

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023