A249247 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	5	6							11
3	17	24	4						45
4	73	112	28						213
5	382	618	192	8					1200
6	2365	3966	1404	104					7839
7	16937	29144	11256	1120	16				58473
8	137861	241848	99384	11872	336				491301
9	1257686	2239650	964000	130480	5280	32			4597128
10	12714721	22914070	10223080	1515440	76880	992			47445183
11	141119617	256809792	117897180	18728320	1111920	21504	64		535688397
12	1706150125	3130061016	1470817380	246701440	16450800	412800	2752		6570596313
13	22321071242	41229842234	19752142704	3461641400	252428960	7611744	78848	128	87024817260
14	314201147513	583760152542	284260937268	51651315800	4043859120	139878816	1932224	7296	1238059230579

Technical details

Node id: n-53575d46e8908019 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-8a820a157fa8f06f

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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:       1;
  1:       2,       2;
  2:       5,       6;
  3:      17,      24,      4;
  4:      73,     112,     28;
  5:     382,     618,    192,      8;
  6:    2365,    3966,   1404,    104;
  7:   16937,   29144,  11256,   1120,   16;
  8:  137861,  241848,  99384,  11872,  336;
  9: 1257686, 2239650, 964000, 130480, 5280, 32;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + n*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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + n*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 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.?
  - 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