A123519 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-9 + 6\,x - 6\,n + 13\,n x + 9\,n^{2} x + 2\,n^{3} x\right)\,P_{n}(x) = \left(-9 - 18\,x + 12\,x^{2} - 6\,n + n x + 22\,n x^{2} + 9\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x + 6\,x^{2} + 6\,x^{3} + 16\,n x^{2} + 8\,n x^{3} + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 12\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	1	12	12								25
1	1	24	60	32							117
2	1	40	180	224	80						525
3	1	60	420	896	720	192					2289
4	1	84	840	2688	3600	2112	448				9773
5	1	112	1512	6720	13200	12672	5824	1024			41065
6	1	144	2520	14784	39600	54912	40768	15360	2304		170393
7	1	180	3960	29568	102960	192192	203840	122880	39168	5120	699869

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-02a0aaf8c813b743 · revision 1 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-27848e5dec2d2c5b

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 ?

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: 1,  12,   12;
  1: 1,  24,   60,    32;
  2: 1,  40,  180,   224,     80;
  3: 1,  60,  420,   896,    720,     192;
  4: 1,  84,  840,  2688,   3600,    2112,     448;
  5: 1, 112, 1512,  6720,  13200,   12672,    5824,    1024;
  6: 1, 144, 2520, 14784,  39600,   54912,   40768,   15360,    2304;
  7: 1, 180, 3960, 29568, 102960,  192192,  203840,  122880,   39168,   5120;
  8: 1, 220, 5940, 54912, 240240,  576576,  815360,  696320,  352512,  97280,  11264;
  9: 1, 264, 8580, 96096, 514800, 1537536, 2772224, 3133440, 2232576, 972800, 236544, 24576;

%F Formula

The row polynomials satisfy -(6*n+9-(2*n^3+9*n^2+13*n+6)*x)*P(n,x) = -(6*n+9-(2*n^3+9*n^2+n-18)*x-(2*n^3+12*n^2+22*n+12)*x^2)*P(n-1,x) - (12*x-(4*n^2+16*n+6)*x^2-(2*n^2+8*n+6)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 12*x^2.

%t Mathematica

P[0, x_] := 1 + 12*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n + 9 - (2*n^3 + 9*n^2 + n - 18)*x - (2*n^3 + 12*n^2 + 22*n + 12)*x^2)*P[n - 1, x] - (12*x - (4*n^2 + 16*n + 6)*x^2 - (2*n^2 + 8*n + 6)*x^3)*D[P[n - 1, x], x])/(-6*n - 9 + (2*n^3 + 9*n^2 + 13*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(6*n + 9 - (2*n^3 + 9*n^2 + n - 18)*x - (2*n^3 + 12*n^2 + 22*n + 12)*x^2)*v[i - 1] - (12*x - (4*n^2 + 16*n + 6)*x^2 - (2*n^2 + 8*n + 6)*x^3)*deriv(v[i - 1], x))/(-6*n - 9 + (2*n^3 + 9*n^2 + 13*n + 6)*x)); 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-5215afd4ce728407
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-5215afd4ce728407
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-5215afd4ce728407
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-5215afd4ce728407

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

preserver_image euler_polynomial, shift 0 e-5215afd4ce728407

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] 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

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023