A123519 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(4 + 6\,x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 11 + 42\,x + 36\,x^{2}\)

\(P_{1}(x) = 41 + 228\,x + 396\,x^{2} + 216\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	11	42	36								89
1	41	228	396	216							881
2	153	1116	2916	3240	1296						8721
3	571	5154	17964	30240	24624	7776					86329
4	2131	22926	99864	225504	278640	178848	46656				854569
5	7953	99336	519048	1470960	2442960	2379456	1259712	279936			8459361
6	29681	422136	2572344	8772624	18318960	23996736	19268928	8678016	1679616		83739041
7	110771	1767294	12303144	49053600	123468624	203521248	219796416	150045696	58786560	10077696	828931049

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-f5286435ed391b4f · revision 1 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-02ef7371f378b284

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): no

OEIS matches ?

A138288: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A072256: 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:      11,       42,        36;
  1:      41,      228,       396,        216;
  2:     153,     1116,      2916,       3240,       1296;
  3:     571,     5154,     17964,      30240,      24624,        7776;
  4:    2131,    22926,     99864,     225504,     278640,      178848,       46656;
  5:    7953,    99336,    519048,    1470960,    2442960,     2379456,     1259712,      279936;
  6:   29681,   422136,   2572344,    8772624,   18318960,    23996736,    19268928,     8678016,     1679616;
  7:  110771,  1767294,  12303144,   49053600,  123468624,   203521248,   219796416,   150045696,    58786560,   10077696;
  8:  413403,  7311666,  57243996,  261260640,  769877136,  1530900000,  2081044224,  1910283264,  1133740800,  393030144,   60466176;
  9: 1542841, 29959788, 260542836, 1339452936, 4523603760, 10539341568, 17289780480, 19977352704, 15937876224, 8364487680, 2600045568, 362797056;

%F Formula

The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 42*x + 36*x^2, P(1,x) = 41 + 228*x + 396*x^2 + 216*x^3.

%t Mathematica

P[0, x_] := 11 + 42*x + 36*x^2;
P[1, x_] := 41 + 228*x + 396*x^2 + 216*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 6*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] = 11 + 42*x + 36*x^2; v[2] = 41 + 228*x + 396*x^2 + 216*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 6*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-8cd26e1d1ff01e6d
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-8cd26e1d1ff01e6d
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-8cd26e1d1ff01e6d
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-8cd26e1d1ff01e6d

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-8cd26e1d1ff01e6d

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8d057b494ddd47e8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-2a4594def0262eef root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023