A038226 → basis A013611

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 32\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 121 + 704\,x + 1024\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	704	1024								1849
1	1331	11616	33792	32768							79507
2	14641	170368	743424	1441792	1048576						3418801
3	161051	2342560	13629440	39649280	57671680	33554432					147008443
4	1771561	30921792	224885760	872284160	1903165440	2214592512	1073741824				6321363049
5	19487171	396829664	3463240704	16791470080	48847912960	85261811712	82678120448	34359738368			271818611107
6	214358881	4988715776	50794196992	295529873408	1074654085120	2501013143552	3637837299712	3023656976384	1099511627776		11688200277601
7	2357947691	61735357728	718375071744	4876242911232	21278150885376	61900075302912	120048630890496	149671020331008	108851651149824	35184372088832	502592611936843

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-a8292da3a4e72c99 · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-e63d418fb4410416

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 ?

A009987: 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,           704,            1024;
  1:         1331,         11616,           33792,            32768;
  2:        14641,        170368,          743424,          1441792,          1048576;
  3:       161051,       2342560,        13629440,         39649280,         57671680,          33554432;
  4:      1771561,      30921792,       224885760,        872284160,       1903165440,        2214592512,        1073741824;
  5:     19487171,     396829664,      3463240704,      16791470080,      48847912960,       85261811712,       82678120448,        34359738368;
  6:    214358881,    4988715776,     50794196992,     295529873408,    1074654085120,     2501013143552,     3637837299712,      3023656976384,      1099511627776;
  7:   2357947691,   61735357728,    718375071744,    4876242911232,   21278150885376,    61900075302912,   120048630890496,    149671020331008,    108851651149824,     35184372088832;
  8:  25937424601,  754543261120,   9877657236480,   76626674319360,  390099432898560,  1361801656664064,  3301337349488640,   5487937412136960,   5986840813240320,   3870280929771520,   1125899906842624;
  9: 285311670611, 9129973459552, 132799613957120, 1158978449080320, 6743147340103680, 27463000076058624, 79892363857625088, 166010106717143040, 241469246134026240, 234151996251176960, 136233888727957504, 36028797018963968;

%F Formula

The row polynomials satisfy P(n,x) = (11+32*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 704*x + 1024*x^2.

%t Mathematica

P[0, x_] := 121 + 704*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 32*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 + 704*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 32*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-376f2b9433b6ff6b
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-376f2b9433b6ff6b
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-376f2b9433b6ff6b

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 A013611 e-376f2b9433b6ff6b

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