Real-rooted atlas

A232535 → 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
\(\left(-9 - 3\,x + 12\,n - n x + 6\,n^{2} x + n^{2} x^{2} - 8\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-9 - 63\,x - 11\,x^{2} - x^{3} + 12\,n + 71\,n x + 23\,n x^{2} - 2\,n x^{3} + 6\,n^{2} x - 11\,n^{2} x^{2} + 9\,n^{2} x^{3} - 8\,n^{3} x - 22\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(24\,x - 22\,x^{2} - 2\,x^{4} - 24\,n x^{2} + 32\,n x^{3} - 8\,n x^{4} - 16\,n^{2} x^{2} + 20\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2116926
31367516128
416429422425608
511008101344525362816
6114418155280445510564912800
71196354916016235951201219116457344
81256630040768929508236828028320081253952
9132410404913922983504084082413325875250491001114112
101400162451860488236501612416146965062016011337376001214849664
11148424255351120202521053721366995534454784014389652048201101114420971520
1215763491462339245426151569004827665596256746241250300730764803506581545616990177536
131676487501052480945587541244060946454601188640008257333531245500615043057408021125196385875968
14178466339170352018500625994593602881119154707014404422855152386020007217710511625120904995282242251644167168

Computed in your browser ?

Technical details
Node id
n-0e8b59f661d3a692 · revision 1 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8e96d4d9e4687e70

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: 1,   4;
  2: 1,  16,     9;
  3: 1,  36,    75,    16;
  4: 1,  64,   294,   224,     25;
  5: 1, 100,   810,  1344,    525,     36;
  6: 1, 144,  1815,  5280,   4455,   1056,     49;
  7: 1, 196,  3549, 16016,  23595,  12012,   1911,    64;
  8: 1, 256,  6300, 40768,  92950,  82368,  28028,  3200,   81;
  9: 1, 324, 10404, 91392, 298350, 408408, 241332, 58752, 5049, 100;
%F Formula
The row polynomials satisfy (12*n-9-(8*n^3-6*n^2+n+3)*x+(2*n^3+n^2)*x^2)*P(n,x) = (12*n-9-(8*n^3-6*n^2-71*n+63)*x-(22*n^3+11*n^2-23*n+11)*x^2+(6*n^3+9*n^2-2*n-1)*x^3)*P(n-1,x) + (24*x-(16*n^2+24*n+22)*x^2+(20*n^2+32*n)*x^3-(4*n^2+8*n+2)*x^4)*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 - 9 - (8*n^3 - 6*n^2 - 71*n + 63)*x - (22*n^3 + 11*n^2 - 23*n + 11)*x^2 + (6*n^3 + 9*n^2 - 2*n - 1)*x^3)*P[n - 1, x] + (24*x - (16*n^2 + 24*n + 22)*x^2 + (20*n^2 + 32*n)*x^3 - (4*n^2 + 8*n + 2)*x^4)*D[P[n - 1, x], x])/(12*n - 9 - (8*n^3 - 6*n^2 + n + 3)*x + (2*n^3 + n^2)*x^2)]];
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 - 9 - (8*n^3 - 6*n^2 - 71*n + 63)*x - (22*n^3 + 11*n^2 - 23*n + 11)*x^2 + (6*n^3 + 9*n^2 - 2*n - 1)*x^3)*v[i - 1] + (24*x - (16*n^2 + 24*n + 22)*x^2 + (20*n^2 + 32*n)*x^3 - (4*n^2 + 8*n + 2)*x^4)*deriv(v[i - 1], x))/(12*n - 9 - (8*n^3 - 6*n^2 + n + 3)*x + (2*n^3 + n^2)*x^2)); 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-e1b53362b0fdfcb4
  • 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-e1b53362b0fdfcb4
  • 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-e1b53362b0fdfcb4
  • 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-e1b53362b0fdfcb4

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