A308737 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 2\,x - x^{2} + n x + n x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 3\,x^{2} - x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	3														7
2	16	22	7													45
3	64	124	76	15												279
4	256	632	564	218	31											1701
5	1024	3056	3552	2024	568	63										10287
6	4096	14304	20400	15280	6360	1398	127									61965
7	16384	65472	110400	102160	56160	18372	3316	255								372519
8	65536	294784	572992	630112	429520	186088	50092	7666	511							2237301
9	262144	1310464	2882560	3668224	2980096	1604512	572992	130960	17392	1023						13430367
10	1048576	5766656	14153472	20442624	19262208	12382272	5503008	1670496	331632	38894	2047					80601885
11	4194304	25164800	68152320	110088960	117949440	88066944	46784640	17691360	4668480	819020	85996	4095				483670359
12	16777216	109049856	322950144	576688640	692017920	588208896	363302016	164349504	54061920	12614360	1982244	188394	8191			2902199301
13	67108864	469757952	1509924864	2952722432	3921561600	3736998144	2629727232	1384061184	544972032	158589200	33159456	4718328	409576	16383		17413727247
14	268435456	2013257728	6979268608	14830899200	21591984128	22791481856	17993234688	10795918848	4948120320	1724342048	449827664	85194512	11075272	884710	32767	104483957805

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-8 - 8\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 3\,x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 4 + 3\,x\)

Technical details

Node id: n-b262f410552e359a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f5047d7bd9492925)
Fingerprint: f6-349ed2f761ce4107

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:      4,       3;
  2:     16,      22,       7;
  3:     64,     124,      76,      15;
  4:    256,     632,     564,     218,      31;
  5:   1024,    3056,    3552,    2024,     568,      63;
  6:   4096,   14304,   20400,   15280,    6360,    1398,    127;
  7:  16384,   65472,  110400,  102160,   56160,   18372,   3316,    255;
  8:  65536,  294784,  572992,  630112,  429520,  186088,  50092,   7666,   511;
  9: 262144, 1310464, 2882560, 3668224, 2980096, 1604512, 572992, 130960, 17392, 1023;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + (n + 2)*x + (n - 1)*x^2)*P[n - 1, x] - (2*x + 3*x^2 + x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4 + (n + 2)*x + (n - 1)*x^2)*v[i - 1] - (2*x + 3*x^2 + x^3)*deriv(v[i - 1], x)); 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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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 of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is t…?
  - ↑ 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