A327997 → x ↦ x+1 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	2														7
2	27	22	4													53
3	159	196	72	8												435
4	1029	1688	932	208	16											3873
5	7353	14898	10900	3640	560	32										37383
6	58095	138366	124816	55080	12640	1440	64									390501
7	506691	1370052	1453508	789992	239120	40544	3584	128								4403619
8	4860297	14548320	17561064	11230240	4183984	934528	122752	8704	256							53450145
9	51023277	165967506	222506280	162155440	71113392	19543776	3378816	355584	20736	512						696065319
10	583097859	2032786998	2972874204	2411972240	1206930800	390607392	83076672	11500800	994560	48640	1024					9693891189
11	7215769575	26678975508	41983498656	37260668808	20794504720	7686659200	1926000384	327997824	37297920	2703360	112640	2048				143914190643
12	96210083853	374213176344	626989080780	600542614320	367404696736	151685512704	43447794368	8720439552	1219941888	116244480	7175168	258048	4096			2270557022337
13	1375803720801	5593601047266	9895834774332	10121818527912	6698362903232	3036743378240	972647515840	223335139456	36837166080	4318718976	350411776	18663424	585728	8192		37959672561255
14	21012273704151	88842112414974	164859653882088	178563241092600	126485386141296	62142371619328	21886678061248	5625230399360	1062161534720	146877739008	14665018368	1026701312	47710208	1318912	16384	670641727353957

Technical details

Node id: n-245fb3a2afc88154 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-5318af082a47bdd8

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:        5,         2;
  2:       27,        22,         4;
  3:      159,       196,        72,         8;
  4:     1029,      1688,       932,       208,       16;
  5:     7353,     14898,     10900,      3640,      560,       32;
  6:    58095,    138366,    124816,     55080,    12640,     1440,      64;
  7:   506691,   1370052,   1453508,    789992,   239120,    40544,    3584,    128;
  8:  4860297,  14548320,  17561064,  11230240,  4183984,   934528,  122752,   8704,   256;
  9: 51023277, 165967506, 222506280, 162155440, 71113392, 19543776, 3378816, 355584, 20736, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(3*n-3)*P(n-2,x) + (n+4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(3*n - 3)*P[n - 2, x] + (n + 4 + 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] = 1; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*n - 3)*v[i - 2] + (n + 4 + 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023