A123097 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} - 4\,n - 4\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 14 + 13\,x + 3\,x^{2}\)

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	14	13	3													30
3	36	52	25	4												117
4	88	172	126	41	5											432
5	208	512	504	248	61	6										1539
6	480	1424	1760	1160	430	85	7									5346
7	1088	3776	5616	4640	2300	684	113	8								18225
8	2432	9664	16800	16688	10360	4116	1022	145	9							61236
9	5376	24064	47872	55552	41440	20608	6832	1456	181	10						203391
10	11776	58624	131328	174336	151872	90720	37632	10704	1998	221	11					669222
11	25600	140288	349440	522240	520320	362880	180768	64320	16020	2660	265	12				2184813
12	55296	330752	906752	1506560	1689600	1347456	783552	334752	104280	23100	3454	313	13			7085880
13	118784	770048	2304000	4212736	5251840	4713984	3134208	1562880	584496	161920	32296	4392	365	14		22851963
14	253952	1773568	5750784	11474944	15741440	15704832	11751168	6699264	2924064	972400	242528	43992	5486	421	15	73338858

Technical details

Node id: n-ac1afc976a1e5fa6 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-b8552aefd92cb15e

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:    5,     2;
  2:   14,    13,     3;
  3:   36,    52,    25,     4;
  4:   88,   172,   126,    41,     5;
  5:  208,   512,   504,   248,    61,     6;
  6:  480,  1424,  1760,  1160,   430,    85,    7;
  7: 1088,  3776,  5616,  4640,  2300,   684,  113,    8;
  8: 2432,  9664, 16800, 16688, 10360,  4116, 1022,  145,   9;
  9: 5376, 24064, 47872, 55552, 41440, 20608, 6832, 1456, 181, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 14 + 13*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[2, x_] := 14 + 13*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*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] = 1; v[2] = 5 + 2*x; v[3] = 14 + 13*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 8 + (2*n - 4)*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)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.?
  - 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

[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