A079510 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	2							4
2	9	15	6						30
3	44	112	92	24					272
4	265	900	1125	610	120				3020
5	1854	7926	13374	11106	4524	720			39504
6	14833	76615	163275	183505	114520	37548	5040		595336
7	133496	809824	2089520	2970080	2509240	1258496	346464	40320	10157440

All 15 rows

Technical details

Node id: n-4770949c8ef98ba1 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-6ce805bb80e27667

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:        2,         2;
  2:        9,        15,         6;
  3:       44,       112,        92,        24;
  4:      265,       900,      1125,       610,        120;
  5:     1854,      7926,     13374,     11106,       4524,       720;
  6:    14833,     76615,    163275,    183505,     114520,     37548,      5040;
  7:   133496,    809824,   2089520,   2970080,    2509240,   1258496,    346464,     40320;
  8:  1334961,   9312912,  28253106,  48657420,   51996105,  35278236,  14827932,   3526416,   362880;
  9: 14684570, 115911930, 404653620, 819701260, 1061328450, 910420770, 517111280, 187411320, 39296160, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (n+1+(n+1)*x)*P(n-2,x) + (n+1+(n+1)*x)*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[(n + 1 + (n + 1)*x)*P[n - 2, x] + (n + 1 + (n + 1)*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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1 + (n + 1)*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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by 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 Triangle T(n,k) read by rows; related to number of preorders.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n,k) read by rows; related to number of preorders.?
  - ↑ Liu–Wang theorem for three-term recurrences (above)

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