Real-rooted atlas

A207606 → basis A007318

nonnegative coefficients

Proved for all n: all roots real, in (−5, −1) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 13 + 11\,x + 2\,x^{2}\)
\(P_{1}(x) = 34 + 44\,x + 17\,x^{2} + 2\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01311226
1344417297
28915593232362
32335104171602921351
4610160816688742453525042
51597492461954130158034841218817
641811476121841177118625258946947270226
71094643540740897084442006160443955608532262087

Show all 15 stored rows

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(\left(18 + 24\,n + 8\,n^{2}\right)\,P_{n}(x) = \left(24 + 12\,x + 28\,n + 14\,n x + 9\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(57 + 45\,x + 6\,x^{2} + 20\,n + 24\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(15 + 18\,x + 3\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 13 + 11\,x + 2\,x^{2}\)
Technical details
Node id
n-dd70177049267f47 · revision 2 · JSON
Definition source
derived:basis_map_A007318
derived:shift_x_plus_one (equal by e-b231262cbab8919e)
Fingerprint
f6-797bb38c8aa87a31

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 ?

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:    13,     11,      2;
  1:    34,     44,     17,      2;
  2:    89,    155,     93,     23,      2;
  3:   233,    510,    417,    160,     29,      2;
  4:   610,   1608,   1668,    874,    245,     35,      2;
  5:  1597,   4924,   6195,   4130,   1580,    348,     41,     2;
  6:  4181,  14761,  21841,  17711,   8625,   2589,    469,    47,    2;
  7: 10946,  43540,  74089,  70844,  42006,  16044,   3955,   608,   53,   2;
  8: 28657, 126805, 243966, 268910, 188237,  87549,  27440,  5732,  765,  59,  2;
  9: 75025, 365532, 784614, 979852, 791615, 434840, 165914, 44028, 7974, 940, 65, 2;
%F Formula
The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+x)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 11*x + 2*x^2, P(1,x) = 34 + 44*x + 17*x^2 + 2*x^3.
%t Mathematica
P[0, x_] := 13 + 11*x + 2*x^2;
P[1, x_] := 34 + 44*x + 17*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 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] = 13 + 11*x + 2*x^2; v[2] = 34 + 44*x + 17*x^2 + 2*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (3 + 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-5ddc9961decc92fb
  • 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-5ddc9961decc92fb, e-a9568ed3b8022fa1
  • 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-5ddc9961decc92fb, e-a9568ed3b8022fa1
  • 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-5ddc9961decc92fb, e-a9568ed3b8022fa1
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-5, -1)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-a9568ed3b8022fa1

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

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