A038333 → 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

\(P_{n}(x) = \left(19 + 21\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 798\,x + 441\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	361	798	441								1600
1	6859	22743	25137	9261							64000
2	130321	576156	955206	703836	194481						2560000
3	2476099	13683705	30248190	33432210	18475695	4084101					102400000
4	47045881	311988474	862073415	1270423980	1053114615	465587514	85766121				4096000000
5	893871739	6915744507	22931152839	42241597335	46688081265	30961569681	11406894093	1801088541			163840000000
6	16983563041	150170452152	580922538588	1284144558984	1774147088070	1568719530504	866923951068	273765458232	37822859361		6553600000000
7	322687697779	3209893414749	14191107728364	36598119931044	60675830411994	67062759929046	49414665210876	23406946678836	6467708950731	794280046581	262144000000000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f96c16e0c5fd8334 · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-7745de9e47db299c

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 ?

A009984: 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:             361,              798,              441;
  1:            6859,            22743,            25137,              9261;
  2:          130321,           576156,           955206,            703836,            194481;
  3:         2476099,         13683705,         30248190,          33432210,          18475695,           4084101;
  4:        47045881,        311988474,        862073415,        1270423980,        1053114615,         465587514,          85766121;
  5:       893871739,       6915744507,      22931152839,       42241597335,       46688081265,       30961569681,       11406894093,        1801088541;
  6:     16983563041,     150170452152,     580922538588,     1284144558984,     1774147088070,     1568719530504,      866923951068,      273765458232,       37822859361;
  7:    322687697779,    3209893414749,   14191107728364,    36598119931044,    60675830411994,    67062759929046,    49414665210876,    23406946678836,     6467708950731,      794280046581;
  8:   6131066257801,   67764416533590,  337038808548645,   993377540985480,  1921401296379810,  2548384877303748,  2347196597516610,  1482439956326280,   614432350319445,   150913208850390,   16679880978201;
  9: 116490258898219, 1416276305552031, 7826790109629645, 25951988258245665, 57367552991911470, 88768739892747222, 98112817776194298, 77457487718048130, 42805453738921335, 15770430324865755, 3486095124444009, 350277500542221;

%F Formula

The row polynomials satisfy P(n,x) = (19+21*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 798*x + 441*x^2.

%t Mathematica

P[0, x_] := 361 + 798*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[(19 + 21*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] = 361 + 798*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 21*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-065a527a7a0e90c1
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-065a527a7a0e90c1
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-065a527a7a0e90c1

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-065a527a7a0e90c1

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-de9af033a7745ebd root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers periodic_product e-400776ae4466efdf 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