A038336 → 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

\(P_{n}(x) = \left(22 + 30\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 484 + 1320\,x + 900\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	484	1320	900								2704
1	10648	43560	59400	27000							140608
2	234256	1277760	2613600	2376000	810000						7311616
3	5153632	35138400	95832000	130680000	89100000	24300000					380204032
4	113379904	927653760	3162456000	5749920000	5880600000	3207600000	729000000				19770609664
5	2494357888	23809779840	97403644800	221371920000	301870800000	246985200000	112266000000	21870000000			1028071702528
6	54875873536	598645893120	2857173580800	7792291584000	13282315200000	14489798400000	9879408000000	3849120000000	656100000000		53459728531456
7	1207269217792	14816485854720	80817195571200	257145622272000	525979681920000	717245020800000	652040928000000	381062880000000	129907800000000	19683000000000	2779905883635712

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-058123003a903c80 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-c62c9860d860e562

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 ?

No OEIS matches.

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:             484,             1320,               900;
  1:           10648,            43560,             59400,              27000;
  2:          234256,          1277760,           2613600,            2376000,             810000;
  3:         5153632,         35138400,          95832000,          130680000,           89100000,            24300000;
  4:       113379904,        927653760,        3162456000,         5749920000,         5880600000,          3207600000,           729000000;
  5:      2494357888,      23809779840,       97403644800,       221371920000,       301870800000,        246985200000,        112266000000,         21870000000;
  6:     54875873536,     598645893120,     2857173580800,      7792291584000,     13282315200000,      14489798400000,       9879408000000,       3849120000000,        656100000000;
  7:   1207269217792,   14816485854720,    80817195571200,    257145622272000,    525979681920000,     717245020800000,     652040928000000,     381062880000000,     129907800000000,     19683000000000;
  8:  26559922791424,  362180765337600,  2222472878208000,   8081719557120000,  19285921670400000,   31558780915200000,   35862251040000000,   27944611200000000,   14289858000000000,   4330260000000000,    590490000000000;
  9: 584318301411328, 8764774521169920, 59759826280704000, 244472016602880000, 666741863462400000, 1272870830246400000, 1735732950336000000, 1690648977600000000, 1152715212000000000, 523961460000000000, 142898580000000000, 17714700000000000;

%F Formula

The row polynomials satisfy P(n,x) = (22+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1320*x + 900*x^2.

%t Mathematica

P[0, x_] := 484 + 1320*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 30*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] = 484 + 1320*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 30*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-022a6c0efe3a8d25
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-022a6c0efe3a8d25
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-022a6c0efe3a8d25

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-022a6c0efe3a8d25

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-2f1e093ae220e593 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-3f0a5f7ceded39cb 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.

A038259 → basis A013610

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