A152842 → x ↦ x+1

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(8 + 10\,x + 3\,x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	3							7
2	8	10	3						21
3	32	64	42	9					147
4	64	160	148	60	9				441
5	256	832	1072	684	216	27			3087
6	512	1920	2976	2440	1116	270	27		9261
7	2048	9216	17664	18688	11784	4428	918	81	64827

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(P_{n}(x) = \left(-6 - 13\,x - 6\,x^{2} + 10\,n + 16\,n x + 6\,n x^{2}\right)\,P_{n-1}(x) + \left(-16 - 36\,x - 26\,x^{2} - 6\,x^{3}\right)\,P_{n-1}'(x)\)

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

Technical details

Node id: n-67db2effba549c79 · revision 2 · JSON
Definition source: derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-9288cdb19cf0727b)
Fingerprint: f6-765e1f4dee9a440f

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     4,     3;
  2:     8,    10,      3;
  3:    32,    64,     42,      9;
  4:    64,   160,    148,     60,      9;
  5:   256,   832,   1072,    684,    216,     27;
  6:   512,  1920,   2976,   2440,   1116,    270,    27;
  7:  2048,  9216,  17664,  18688,  11784,   4428,   918,    81;
  8:  4096, 20480,  44544,  55040,  42256,  20640,  6264,  1080,   81;
  9: 16384, 94208, 239616, 353792, 334144, 209328, 86976, 23112, 3564, 243;

%F Formula

The row polynomials satisfy P(n,x) = (8+10*x+3*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 10*x + 3*x^2)*P[n - 2, 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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 10*x + 3*x^2)*v[i - 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-56752775b3d97dc0
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-1c523c2967bf00ac, e-56752775b3d97dc0
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-1c523c2967bf00ac, e-56752775b3d97dc0
Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-1c523c2967bf00ac

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

basis_map A007318 e-56752775b3d97dc0

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-596b986e3a00db44 root: theorem check
    
    Parent-row provers of polytool polytool ?
definition_equality e-9288cdb19cf0727b root: theorem check

Ore-algebra identity of two definitions polytool ?
preserver_image affine, shift 0 e-1c523c2967bf00ac

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DEL…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ parent_provers periodic_product e-596b986e3a00db44 (expanded above)

Related sequences ?

No candidate equalities or related sequences.