A038334 → 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

\(P_{n}(x) = \left(20 + 8\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 320\,x + 64\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	400	320	64								784
1	8000	9600	3840	512							21952
2	160000	256000	153600	40960	4096						614656
3	3200000	6400000	5120000	2048000	409600	32768					17210368
4	64000000	153600000	153600000	81920000	24576000	3932160	262144				481890304
5	1280000000	3584000000	4300800000	2867200000	1146880000	275251200	36700160	2097152			13492928512
6	25600000000	81920000000	114688000000	91750400000	45875200000	14680064000	2936012800	335544320	16777216		377801998336
7	512000000000	1843200000000	2949120000000	2752512000000	1651507200000	660602880000	176160768000	30198988800	3019898880	134217728	10578455953408

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-64a6f5553b7b7b7f · revision 1 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-9629d1e9c04dbfbe

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 ?

A009972: 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:             400,             320,               64;
  1:            8000,            9600,             3840,              512;
  2:          160000,          256000,           153600,            40960,             4096;
  3:         3200000,         6400000,          5120000,          2048000,           409600,           32768;
  4:        64000000,       153600000,        153600000,         81920000,         24576000,         3932160,          262144;
  5:      1280000000,      3584000000,       4300800000,       2867200000,       1146880000,       275251200,        36700160,         2097152;
  6:     25600000000,     81920000000,     114688000000,      91750400000,      45875200000,     14680064000,      2936012800,       335544320,       16777216;
  7:    512000000000,   1843200000000,    2949120000000,    2752512000000,    1651507200000,    660602880000,    176160768000,     30198988800,     3019898880,     134217728;
  8:  10240000000000,  40960000000000,   73728000000000,   78643200000000,   55050240000000,  26424115200000,   8808038400000,   2013265920000,   301989888000,   26843545600,   1073741824;
  9: 204800000000000, 901120000000000, 1802240000000000, 2162688000000000, 1730150400000000, 968884224000000, 387553689600000, 110729625600000, 22145925120000, 2952790016000, 236223201280, 8589934592;

%F Formula

The row polynomials satisfy P(n,x) = (20+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 320*x + 64*x^2.

%t Mathematica

P[0, x_] := 400 + 320*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 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] = 400 + 320*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 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-ec87a016e71e40fc
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-ec87a016e71e40fc
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-ec87a016e71e40fc

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-ec87a016e71e40fc

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-f4f0736bd14a35db root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-b445a5ff4f115ce0 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (6)

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