A038301 → 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

\(P_{n}(x) = \left(20 + 33\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 400 + 1320\,x + 1089\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	400	1320	1089								2809
1	8000	39600	65340	35937							148877
2	160000	1056000	2613600	2874960	1185921						7890481
3	3200000	26400000	87120000	143748000	118592100	39135393					418195493
4	64000000	633600000	2613600000	5749920000	7115526000	4696247160	1291467969				22164361129
5	1280000000	14784000000	73180800000	201247200000	332057880000	328737301200	180805515660	42618442977			1174711139837
6	25600000000	337920000000	1951488000000	6439910400000	13282315200000	17532656064000	14464441252800	6818950876320	1406408618241		62259690411361
7	512000000000	7603200000000	50181120000000	193197312000000	478163347200000	788969522880000	867866475168000	613705578868800	253153551283380	46411484401953	3299763591802133

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-82575db1f46b862e · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-72a6360b947e7167

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:             400,             1320,              1089;
  1:            8000,            39600,             65340,              35937;
  2:          160000,          1056000,           2613600,            2874960,            1185921;
  3:         3200000,         26400000,          87120000,          143748000,          118592100,            39135393;
  4:        64000000,        633600000,        2613600000,         5749920000,         7115526000,          4696247160,          1291467969;
  5:      1280000000,      14784000000,       73180800000,       201247200000,       332057880000,        328737301200,        180805515660,         42618442977;
  6:     25600000000,     337920000000,     1951488000000,      6439910400000,     13282315200000,      17532656064000,      14464441252800,       6818950876320,       1406408618241;
  7:    512000000000,    7603200000000,    50181120000000,    193197312000000,    478163347200000,     788969522880000,     867866475168000,     613705578868800,     253153551283380,      46411484401953;
  8:  10240000000000,  168960000000000,  1254528000000000,   5519923200000000,  15938778240000000,   31558780915200000,   43393323758400000,   40913705257920000,   25315355128338000,    9282296880390600,   1531578985264449;
  9: 204800000000000, 3717120000000000, 30666240000000000, 151797888000000000, 500933030400000000, 1157155300224000000, 1909306245369600000, 2250253789185600000, 1856459376078120000, 1021052656842966000, 336947376758178780, 50542106513726817;

%F Formula

The row polynomials satisfy P(n,x) = (20+33*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1320*x + 1089*x^2.

%t Mathematica

P[0, x_] := 400 + 1320*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 33*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 + 1320*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 33*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-73a1a035b42069f4
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-73a1a035b42069f4
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-73a1a035b42069f4

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-73a1a035b42069f4

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-c85fc2223c79991e root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-38b35fa9e37eb25d root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.

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