A038226 → basis A007318

nonnegative coefficients only rational roots

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

Recurrence

\(P_{n}(x) = \left(11 + 8\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 121 + 176\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	121	176	64								361
1	1331	2904	2112	512							6859
2	14641	42592	46464	22528	4096						130321
3	161051	585640	851840	619520	225280	32768					2476099
4	1771561	7730448	14055360	13629440	7434240	2162688	262144				47045881
5	19487171	99207416	216452544	262366720	190812160	83263488	20185088	2097152			893871739
6	214358881	1247178944	3174637312	4617654272	4197867520	2442395648	888143872	184549376	16777216		16983563041
7	2357947691	15433839432	44898441984	76191295488	83117776896	60449292288	29308747776	9135194112	1660944384	134217728	322687697779

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2e0a8214849adcd6 · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-e348a50f4e54dede

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 ?

A001029: 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,           176,            64;
  1:         1331,          2904,          2112,            512;
  2:        14641,         42592,         46464,          22528,           4096;
  3:       161051,        585640,        851840,         619520,         225280,          32768;
  4:      1771561,       7730448,      14055360,       13629440,        7434240,        2162688,         262144;
  5:     19487171,      99207416,     216452544,      262366720,      190812160,       83263488,       20185088,        2097152;
  6:    214358881,    1247178944,    3174637312,     4617654272,     4197867520,     2442395648,      888143872,      184549376,      16777216;
  7:   2357947691,   15433839432,   44898441984,    76191295488,    83117776896,    60449292288,    29308747776,     9135194112,    1660944384,    134217728;
  8:  25937424601,  188635815280,  617353577280,  1197291786240,  1523825909760,  1329884430336,   805990563840,   334957117440,   91351941120,  14763950080,   1073741824;
  9: 285311670611, 2282493364888, 8299975872320, 18109038266880, 26340419297280, 26819336011776, 19504971644928, 10132452802560, 3684528291840, 893218979840, 129922760704, 8589934592;

%F Formula

The row polynomials satisfy P(n,x) = (11+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 176*x + 64*x^2.

%t Mathematica

P[0, x_] := 121 + 176*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 8*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 + 176*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 8*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-0566c972c7be0ca0
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-0566c972c7be0ca0
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-0566c972c7be0ca0

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 A007318 e-0566c972c7be0ca0

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.

A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.

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