A155112 → basis A007318

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 3 + 4\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	3	4	1													8
3	8	14	7	1												30
4	22	47	34	10	1											114
5	60	152	143	63	13	1										432
6	164	480	553	323	101	16	1									1638
7	448	1488	2024	1468	614	148	19	1								6210
8	1224	4548	7122	6159	3221	1043	204	22	1							23544
9	3344	13744	24328	24400	15297	6217	1637	269	25	1						89262
10	9136	41152	81192	92568	67595	33038	10942	2423	343	28	1					338418
11	24960	122272	265936	339456	282752	161402	64413	17963	3428	426	31	1				1283040
12	68192	360944	857680	1211176	1132718	739227	345150	116127	27928	4679	518	34	1			4864374
13	186304	1059584	2730448	4224880	4381572	3216728	1719755	677743	196802	41566	6203	619	37	1		18442242
14	508992	3095552	8596784	14460240	16464636	13426200	8085765	3652645	1243330	317220	59687	8027	729	40	1	69919848

Technical details

Node id: n-44b7e92938a6ed1b · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-080d08fa21c15dd2

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:    1,     1;
  2:    3,     4,     1;
  3:    8,    14,     7,     1;
  4:   22,    47,    34,    10,     1;
  5:   60,   152,   143,    63,    13,    1;
  6:  164,   480,   553,   323,   101,   16,    1;
  7:  448,  1488,  2024,  1468,   614,  148,   19,   1;
  8: 1224,  4548,  7122,  6159,  3221, 1043,  204,  22,  1;
  9: 3344, 13744, 24328, 24400, 15297, 6217, 1637, 269, 25, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*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] = 1; v[2] = 1 + x; v[3] = 3 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (n - 2)*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), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where…?
  - 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