A038226 → basis A013609

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence

\(P_{n}(x) = \left(11 + 16\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 121 + 352\,x + 256\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	352	256								729
1	1331	5808	8448	4096							19683
2	14641	85184	185856	180224	65536						531441
3	161051	1171280	3407360	4956160	3604480	1048576					14348907
4	1771561	15460896	56221440	109035520	118947840	69206016	16777216				387420489
5	19487171	198414832	865810176	2098933760	3052994560	2664431616	1291845632	268435456			10460353203
6	214358881	2494357888	12698549248	36941234176	67165880320	78156660736	56841207808	23622320128	4294967296		282429536481
7	2357947691	30867678864	179593767936	609530363904	1329884430336	1934377353216	1875759857664	1169304846336	425201762304	68719476736	7625597484987

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-1600c9e1b40ceecc · revision 1 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-89a932e93fe55d02

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 ?

A009971: 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:          121,           352,            256;
  1:         1331,          5808,           8448,            4096;
  2:        14641,         85184,         185856,          180224,           65536;
  3:       161051,       1171280,        3407360,         4956160,         3604480,         1048576;
  4:      1771561,      15460896,       56221440,       109035520,       118947840,        69206016,         16777216;
  5:     19487171,     198414832,      865810176,      2098933760,      3052994560,      2664431616,       1291845632,        268435456;
  6:    214358881,    2494357888,    12698549248,     36941234176,     67165880320,     78156660736,      56841207808,      23622320128,      4294967296;
  7:   2357947691,   30867678864,   179593767936,    609530363904,   1329884430336,   1934377353216,    1875759857664,    1169304846336,    425201762304,     68719476736;
  8:  25937424601,  377271630560,  2469414309120,   9578334289920,  24381214556160,  42556301770752,   51583396085760,   42874511032320,  23386096926720,   7559142440960,   1099511627776;
  9: 285311670611, 4564986729776, 33199903489280, 144872306135040, 421446708756480, 858218752376832, 1248318185275392, 1296953958727680, 943239242711040, 457328117678080, 133040906960896, 17592186044416;

%F Formula

The row polynomials satisfy P(n,x) = (11+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 352*x + 256*x^2.

%t Mathematica

P[0, x_] := 121 + 352*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 16*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] = 121 + 352*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 16*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, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c8a2ee9cf24a53ed
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-c8a2ee9cf24a53ed
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-c8a2ee9cf24a53ed

Proof chains

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

basis_map A013609 e-c8a2ee9cf24a53ed

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-fdf6b848b46b3343 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-711ca5eff5ba56ea root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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