A038281 → 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

\(P_{n}(x) = \left(11 + 12\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 121 + 264\,x + 144\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	264	144								529
1	1331	4356	4752	1728							12167
2	14641	63888	104544	76032	20736						279841
3	161051	878460	1916640	2090880	1140480	248832					6436343
4	1771561	11595672	31624560	45999360	37635840	16422912	2985984				148035889
5	19487171	148811124	487018224	885487680	965986560	632282112	229920768	35831808			3404825447
6	214358881	1870768416	7142933952	15584583168	21251704320	18546941952	10116513792	3153199104	429981696		78310985281
7	2357947691	23150759148	101021494464	257145622272	420783745536	459036813312	333844955136	156083355648	42568187904	5159780352	1801152661463

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f55a7ba0b32633bc · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-60cc3cc9b4461a16

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 ?

A009967: 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,           264,            144;
  1:         1331,          4356,           4752,           1728;
  2:        14641,         63888,         104544,          76032,           20736;
  3:       161051,        878460,        1916640,        2090880,         1140480,          248832;
  4:      1771561,      11595672,       31624560,       45999360,        37635840,        16422912,         2985984;
  5:     19487171,     148811124,      487018224,      885487680,       965986560,       632282112,       229920768,        35831808;
  6:    214358881,    1870768416,     7142933952,    15584583168,     21251704320,     18546941952,     10116513792,      3153199104,      429981696;
  7:   2357947691,   23150759148,   101021494464,   257145622272,    420783745536,    459036813312,    333844955136,    156083355648,    42568187904,     5159780352;
  8:  25937424601,  282953722920,  1389045548880,  4040859778560,   7714368668160,  10098809892864,   9180736266240,   5723056373760,  2341250334720,   567575838720,   61917364224;
  9: 285311670611, 3423740047332, 18674945712720, 61118004150720, 133348372692480, 203659332839424, 222173817643008, 173122455306240, 94430430167040, 34338338242560, 7492001071104, 743008370688;

%F Formula

The row polynomials satisfy P(n,x) = (11+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 264*x + 144*x^2.

%t Mathematica

P[0, x_] := 121 + 264*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 12*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 + 264*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 12*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-8f18b93eb5dfe7b3
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-8f18b93eb5dfe7b3
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-8f18b93eb5dfe7b3

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-8f18b93eb5dfe7b3

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-da334b7c58401d78 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-c5656743efdffc45 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

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