Real-rooted atlas

A103327 → (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
\(\left(-1 + x + 2\,n - 11\,n x + 28\,n^{2} x - n^{2} x^{2} - 20\,n^{3} x + 6\,n^{3} x^{2} - 12\,n^{4} x^{2} + 8\,n^{5} x^{2}\right)\,P_{n}(x) = \left(1 - 21\,x - 37\,x^{2} - 3\,x^{3} + 2\,n + 31\,n x + 89\,n x^{2} + 14\,n x^{3} + 8\,n^{2} x + 49\,n^{2} x^{2} - 13\,n^{2} x^{3} - 20\,n^{3} x - 110\,n^{3} x^{2} - 38\,n^{3} x^{3} - 4\,n^{4} x^{2} + 20\,n^{4} x^{3} + 8\,n^{5} x^{2} + 24\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 10\,x^{2} - 12\,x^{3} - 2\,x^{4} - 24\,n x^{2} + 16\,n x^{3} + 8\,n x^{4} - 40\,n^{2} x^{2} + 44\,n^{2} x^{3} - 4\,n^{2} x^{4} + 32\,n^{3} x^{3} - 32\,n^{3} x^{4} + 16\,n^{4} x^{3} - 16\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
2540954
3714018916352
493361134576252080
5116604158528013753611520
613114411583274561787528084960928
71518202702710296012512549140514564311296

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-dadfb2332eb41729 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2a9ba3be646fcc5b

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:  5,   40,      9;
  3:  7,  140,    189,     16;
  4:  9,  336,   1134,    576,      25;
  5: 11,  660,   4158,   5280,    1375,      36;
  6: 13, 1144,  11583,  27456,   17875,    2808,      49;
  7: 15, 1820,  27027, 102960,  125125,   49140,    5145,     64;
  8: 17, 2720,  55692, 311168,  607750,  445536,  116620,   8704,    81;
  9: 19, 3876, 104652, 806208, 2309450, 2720952, 1329468, 248064, 13851, 100;
%F Formula
The row polynomials satisfy (2*n-1-(20*n^3-28*n^2+11*n-1)*x+(8*n^5-12*n^4+6*n^3-n^2)*x^2)*P(n,x) = (2*n+1-(20*n^3-8*n^2-31*n+21)*x+(8*n^5-4*n^4-110*n^3+49*n^2+89*n-37)*x^2+(24*n^5+20*n^4-38*n^3-13*n^2+14*n-3)*x^3)*P(n-1,x) + (4*x-(40*n^2+24*n-10)*x^2+(16*n^4+32*n^3+44*n^2+16*n-12)*x^3-(16*n^4+32*n^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[((2*n + 1 - (20*n^3 - 8*n^2 - 31*n + 21)*x + (8*n^5 - 4*n^4 - 110*n^3 + 49*n^2 + 89*n - 37)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*P[n - 1, x] + (4*x - (40*n^2 + 24*n - 10)*x^2 + (16*n^4 + 32*n^3 + 44*n^2 + 16*n - 12)*x^3 - (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*D[P[n - 1, x], x])/(2*n - 1 - (20*n^3 - 28*n^2 + 11*n - 1)*x + (8*n^5 - 12*n^4 + 6*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] = ((2*n + 1 - (20*n^3 - 8*n^2 - 31*n + 21)*x + (8*n^5 - 4*n^4 - 110*n^3 + 49*n^2 + 89*n - 37)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*v[i - 1] + (4*x - (40*n^2 + 24*n - 10)*x^2 + (16*n^4 + 32*n^3 + 44*n^2 + 16*n - 12)*x^3 - (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*deriv(v[i - 1], x))/(2*n - 1 - (20*n^3 - 28*n^2 + 11*n - 1)*x + (8*n^5 - 12*n^4 + 6*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-a07ebdda950bedd8
  • 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-a07ebdda950bedd8
  • 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-a07ebdda950bedd8
  • 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-a07ebdda950bedd8

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