A159854 → 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

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

\(P_{0}(x) = 1 + 2\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	2	1															4
1	2	5	4	1														12
2	4	12	13	6	1													36
3	8	28	38	25	8	1												108
4	16	64	104	88	41	10	1											324
5	32	144	272	280	170	61	12	1										972
6	64	320	688	832	620	292	85	14	1									2916
7	128	704	1696	2352	2072	1204	462	113	16	1								8748
8	256	1536	4096	6400	6496	4480	2128	688	145	18	1							26244
9	512	3328	9728	16896	19392	15456	8736	3504	978	181	20	1						78732
10	1024	7168	22784	43520	55680	50304	32928	15744	5460	1340	221	22	1					236196
11	2048	15360	52736	109824	154880	156288	116160	64416	26664	8140	1782	265	24	1				708588
12	4096	32768	120832	272384	419584	467456	388608	244992	117744	42944	11704	2312	313	26	1			2125764
13	8192	69632	274432	665600	1111552	1354496	1244672	878592	480480	203632	66352	16328	2938	365	28	1		6377292
14	16384	147456	618496	1605632	2888704	3820544	3843840	3001856	1839552	887744	336336	99008	22204	3668	421	30	1	19131876

Computed in your browser ?

Technical details

Node id: n-21d2f90cb223c8f5 · revision 1 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-a3dcade816dfaa44

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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 ?

A168681: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168777: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A052156: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A167882: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168825: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A025579: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A003946: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A168729: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    2,    1;
  1:   2,    5,    4,     1;
  2:   4,   12,   13,     6,     1;
  3:   8,   28,   38,    25,     8,     1;
  4:  16,   64,  104,    88,    41,    10,    1;
  5:  32,  144,  272,   280,   170,    61,   12,    1;
  6:  64,  320,  688,   832,   620,   292,   85,   14,   1;
  7: 128,  704, 1696,  2352,  2072,  1204,  462,  113,  16,   1;
  8: 256, 1536, 4096,  6400,  6496,  4480, 2128,  688, 145,  18,  1;
  9: 512, 3328, 9728, 16896, 19392, 15456, 8736, 3504, 978, 181, 20, 1;

%F Formula

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

%t Mathematica

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

Generated with polytool c48629374597.

Proved properties

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-34caf7309ed0b453
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-34caf7309ed0b453, e-7a44d9ed78450a4e
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-34caf7309ed0b453, e-7a44d9ed78450a4e
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-7a44d9ed78450a4e

Proof chains

2 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-7a44d9ed78450a4e

Geometric basis map (elementary) polytool ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers periodic_product e-b62074ba45057e82 root: theorem check
    
    Parent-row provers of polytool polytool ?
preserver_image affine, shift 0 e-34caf7309ed0b453

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1-x,x/(1-x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - ↵ parent_provers periodic_product e-b62074ba45057e82 (expanded above)

Related sequences ?

No candidate equalities or related sequences.