Real-rooted atlas

A110441 → (θ+1)²

nonnegative coefficients

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

Recurrence
\(\left(8\,n + 6\,n x + 18\,n x^{2} - 20\,n^{3} x^{2} - 30\,n^{3} x^{3} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(12 + 18\,x + 42\,x^{2} + 30\,x^{3} - 6\,x^{4} + 12\,n + 25\,n x + 39\,n x^{2} + 13\,n x^{3} - 33\,n x^{4} + 7\,n^{2} x - 33\,n^{2} x^{2} - 77\,n^{2} x^{3} - 54\,n^{2} x^{4} + n^{2} x^{5} - 30\,n^{3} x^{2} - 62\,n^{3} x^{3} - 27\,n^{3} x^{4} + 3\,n^{3} x^{5} - 2\,n^{4} x^{3} + 3\,n^{4} x^{4} + 3\,n^{4} x^{5} + 3\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(1 + 9\,x + 26\,x^{2} + 42\,x^{3} - 11\,x^{4} - 3\,x^{5} - 7\,n x^{2} - 57\,n x^{3} - 97\,n x^{4} - 15\,n x^{5} - 7\,n^{2} x^{2} - 57\,n^{2} x^{3} - 96\,n^{2} x^{4} - 9\,n^{2} x^{5} + n^{2} x^{6} + 2\,n^{3} x^{4} + 12\,n^{3} x^{5} + 2\,n^{3} x^{6} + n^{4} x^{4} + 6\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
2724940
315928116204
43128843219225968
56380417821312375364372
612720886291672031256484919048
72555164200072872019125637210296480736

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-8d7b2a6627d99246 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-0cd5895e164db44e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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;
  1:    3,     4;
  2:    7,    24,      9;
  3:   15,    92,     81,     16;
  4:   31,   288,    432,    192,     25;
  5:   63,   804,   1782,   1312,    375,     36;
  6:  127,  2088,   6291,   6720,   3125,    648,    49;
  7:  255,  5164,  20007,  28720,  19125,   6372,  1029,    64;
  8:  511, 12336,  59058, 108288,  96000,  45360, 11662,  1536,   81;
  9: 1023, 28724, 164916, 372416, 418950, 261576, 94668, 19712, 2187, 100;
%F Formula
The row polynomials satisfy (8*n+6*n*x-(20*n^3-18*n)*x^2-30*n^3*x^3+2*n^5*x^4)*P(n,x) = (12*n+12+(7*n^2+25*n+18)*x-(30*n^3+33*n^2-39*n-42)*x^2-(2*n^4+62*n^3+77*n^2-13*n-30)*x^3+(3*n^5+3*n^4-27*n^3-54*n^2-33*n-6)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (1+9*x-(7*n^2+7*n-26)*x^2-(57*n^2+57*n-42)*x^3+(n^4+2*n^3-96*n^2-97*n-11)*x^4+(6*n^4+12*n^3-9*n^2-15*n-3)*x^5+(n^4+2*n^3+n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 12 + (7*n^2 + 25*n + 18)*x - (30*n^3 + 33*n^2 - 39*n - 42)*x^2 - (2*n^4 + 62*n^3 + 77*n^2 - 13*n - 30)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (1 + 9*x - (7*n^2 + 7*n - 26)*x^2 - (57*n^2 + 57*n - 42)*x^3 + (n^4 + 2*n^3 - 96*n^2 - 97*n - 11)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(8*n + 6*n*x - (20*n^3 - 18*n)*x^2 - 30*n^3*x^3 + 2*n^5*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 12 + (7*n^2 + 25*n + 18)*x - (30*n^3 + 33*n^2 - 39*n - 42)*x^2 - (2*n^4 + 62*n^3 + 77*n^2 - 13*n - 30)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (1 + 9*x - (7*n^2 + 7*n - 26)*x^2 - (57*n^2 + 57*n - 42)*x^3 + (n^4 + 2*n^3 - 96*n^2 - 97*n - 11)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(8*n + 6*n*x - (20*n^3 - 18*n)*x^2 - 30*n^3*x^3 + 2*n^5*x^4)); 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-f01a2d79bc5e0611
  • 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-f01a2d79bc5e0611
  • 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-f01a2d79bc5e0611
  • 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-f01a2d79bc5e0611

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

No candidate equalities or related sequences.

References

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