A153648 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-30 - 30\,x + 15\,n + 15\,n x\right)\,P_{n-2}(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 10 + 5\,x\)

\(P_{2}(x) = 50 + 50\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	10	5														15
2	50	50	2													102
3	250	375	129	2												756
4	1250	2500	1413	163	2											5328
5	6250	15625	12886	3704	197	2										38664
6	31250	93750	100092	43934	6573	231	2									275832
7	156250	546875	721599	436810	115595	10020	265	2								1987416
8	781250	3125000	4897703	3755609	1425605	237695	14045	299	2							14237208
9	3906250	17578125	31947516	29785056	14892894	3785220	420060	18648	333	2						102334104
10	19531250	97656250	201813702	221317308	137289054	47412834	8401440	672516	23829	367	2					734118552
11	97656250	537109375	1244168269	1570436898	1166064666	512286432	127294914	16327092	1004889	29588	401	2				5272378776
12	488281250	2929687500	7517495193	10732007215	9281661660	4961165850	1604090370	296058438	28782045	1427005	35925	435	2			37840692888
13	2441406250	15869140625	44683842546	71200587128	70342853995	44379259830	17763066780	4350537336	613602243	47153210	1948690	42840	469	2		271693441944
14	12207031250	85449218750	261944566112	460827552922	512091328243	372743786305	178384236690	54966373572	10450349067	1161044413	72994540	2579770	50333	503	2	1950301112472

Technical details

Node id: n-ec2cafb410ca50bb · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-5d7a6ea0349af6c7

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:       2;
  1:      10,        5;
  2:      50,       50,        2;
  3:     250,      375,      129,        2;
  4:    1250,     2500,     1413,      163,        2;
  5:    6250,    15625,    12886,     3704,      197,       2;
  6:   31250,    93750,   100092,    43934,     6573,     231,      2;
  7:  156250,   546875,   721599,   436810,   115595,   10020,    265,     2;
  8:  781250,  3125000,  4897703,  3755609,  1425605,  237695,  14045,   299,   2;
  9: 3906250, 17578125, 31947516, 29785056, 14892894, 3785220, 420060, 18648, 333, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (15*n-30+(15*n-30)*x)*P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 10 + 5*x, P(2,x) = 50 + 50*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 10 + 5*x;
P[2, x_] := 50 + 50*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15*n - 30 + (15*n - 30)*x)*P[n - 2, x] + (2*n - 4 + (n - 2)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 5*x; v[3] = 50 + 50*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((15*n - 30 + (15*n - 30)*x)*v[i - 2] + (2*n - 4 + (n - 2)*x)*v[i - 1])/(n - 2)); 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 ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check

References

[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