A038226 → basis A013610

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 24\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 121 + 528\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	528	576								1225
1	1331	8712	19008	13824							42875
2	14641	127776	418176	608256	331776						1500625
3	161051	1756920	7666560	16727040	18247680	7962624					52521875
4	1771561	23191344	126498240	367994880	602173440	525533184	191102976				1838265625
5	19487171	297622248	1948072896	7083901440	15455784960	20233027584	14714929152	4586471424			64339296875
6	214358881	3741536832	28571735808	124676665344	340027269120	593502142464	647456882688	403609485312	110075314176		2251875390625
7	2357947691	46301518296	404085977856	2057164978176	6732539928576	14689178025984	21366077128704	19978669522944	10897456103424	2641807540224	78815638671875

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-40f57961a3f2ab5d · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-83fc6524b6311133

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 ?

A009979: 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,           528,            576;
  1:         1331,          8712,          19008,           13824;
  2:        14641,        127776,         418176,          608256,           331776;
  3:       161051,       1756920,        7666560,        16727040,         18247680,          7962624;
  4:      1771561,      23191344,      126498240,       367994880,        602173440,        525533184,         191102976;
  5:     19487171,     297622248,     1948072896,      7083901440,      15455784960,      20233027584,       14714929152,        4586471424;
  6:    214358881,    3741536832,    28571735808,    124676665344,     340027269120,     593502142464,      647456882688,      403609485312,      110075314176;
  7:   2357947691,   46301518296,   404085977856,   2057164978176,    6732539928576,   14689178025984,    21366077128704,    19978669522944,    10897456103424,     2641807540224;
  8:  25937424601,  565907445840,  5556182195520,  32326878228480,  123429898690560,  323161916571648,   587567121039360,   732551215841280,   599360085688320,   290598829424640,   63403380965376;
  9: 285311670611, 6847480094664, 74699782850880, 488944033205760, 2133573963079680, 6517098650861568, 14219124329152512, 22159674279198720, 24174190122762240, 17581229180190720, 7671809096810496, 1521681143169024;

%F Formula

The row polynomials satisfy P(n,x) = (11+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 528*x + 576*x^2.

%t Mathematica

P[0, x_] := 121 + 528*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 24*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 + 528*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 24*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-edadd00b11ba8928
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-edadd00b11ba8928
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-edadd00b11ba8928

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 A013610 e-edadd00b11ba8928

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 ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A038314 → basis A013611

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