Real-rooted atlas

A126216 → (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
\(\left(8 - 4\,x + 20\,n + 16\,n^{2} + 19\,n^{2} x + 2\,n^{2} x^{2} + 4\,n^{3} + 21\,n^{3} x + 5\,n^{3} x^{2} + 6\,n^{4} x + 4\,n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(8 - 4\,x + 32\,n + 42\,n x + 2\,n x^{3} + 40\,n^{2} + 118\,n^{2} x + 58\,n^{2} x^{2} + 9\,n^{2} x^{3} + 16\,n^{3} + 96\,n^{3} x + 90\,n^{3} x^{2} + 20\,n^{3} x^{3} + 24\,n^{4} x + 40\,n^{4} x^{2} + 14\,n^{4} x^{3} + 4\,n^{5} x^{2} + 3\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 14\,x^{2} - 8\,x^{3} - 2\,x^{4} - 16\,n x - 47\,n x^{2} - 40\,n x^{3} - 9\,n x^{4} - 8\,n^{2} x - 44\,n^{2} x^{2} - 52\,n^{2} x^{3} - 16\,n^{2} x^{4} - 12\,n^{3} x^{2} - 22\,n^{3} x^{3} - 10\,n^{3} x^{4} - 2\,n^{4} x^{3} - 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
2520934
314848116195
442336504224251131
5132132027001920500366608
64295148133651320056259724938788
714302002063063800804812513860171564228357
8486277792288288448448350350144144301842816811346965
91679630232812888722376192229320012383283745565990443741007954650
105878611757205668650120931201392300093562563745560873600110565650012147011878
11208012457626424622290596904007994250064465632323270641005312018711001925009317144277958343
127429001782960010590782428765088043968375041442192025069968098233344245381403740000319440129601691643780607
132674440695354404519803601359804160233716340025241364721792502712851355648269374248554268007059624509184175761969721540260
1496948452714556601916570565632772504012084197125147241294201202470569367310305922592727137678978300117366249126977767843292332422557492086280

Computed in your browser ?

Technical details
Node id
n-d2911c15cf811244 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-097fa7332775e3f8

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:     2,      4;
  2:     5,     20,       9;
  3:    14,     84,      81,      16;
  4:    42,    336,     504,     224,      25;
  5:   132,   1320,    2700,    1920,     500,      36;
  6:   429,   5148,   13365,   13200,    5625,     972,     49;
  7:  1430,  20020,   63063,   80080,   48125,   13860,   1715,    64;
  8:  4862,  77792,  288288,  448448,  350350,  144144,  30184,  2816,   81;
  9: 16796, 302328, 1288872, 2376192, 2293200, 1238328, 374556, 59904, 4374, 100;
%F Formula
The row polynomials satisfy (4*n^3+16*n^2+20*n+8+(6*n^4+21*n^3+19*n^2-4)*x+(n^5+4*n^4+5*n^3+2*n^2)*x^2)*P(n,x) = (16*n^3+40*n^2+32*n+8+(24*n^4+96*n^3+118*n^2+42*n-4)*x+(4*n^5+40*n^4+90*n^3+58*n^2)*x^2+(3*n^5+14*n^4+20*n^3+9*n^2+2*n)*x^3)*P(n-1,x) - ((8*n^2+16*n+8)*x+(12*n^3+44*n^2+47*n+14)*x^2+(2*n^4+22*n^3+52*n^2+40*n+8)*x^3+(2*n^4+10*n^3+16*n^2+9*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[((16*n^3 + 40*n^2 + 32*n + 8 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (4*n^5 + 40*n^4 + 90*n^3 + 58*n^2)*x^2 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^3)*P[n - 1, x] - ((8*n^2 + 16*n + 8)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (2*n^4 + 22*n^3 + 52*n^2 + 40*n + 8)*x^3 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 16*n^2 + 20*n + 8 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (n^5 + 4*n^4 + 5*n^3 + 2*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] = ((16*n^3 + 40*n^2 + 32*n + 8 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (4*n^5 + 40*n^4 + 90*n^3 + 58*n^2)*x^2 + (3*n^5 + 14*n^4 + 20*n^3 + 9*n^2 + 2*n)*x^3)*v[i - 1] - ((8*n^2 + 16*n + 8)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (2*n^4 + 22*n^3 + 52*n^2 + 40*n + 8)*x^3 + (2*n^4 + 10*n^3 + 16*n^2 + 9*n + 2)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 16*n^2 + 20*n + 8 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (n^5 + 4*n^4 + 5*n^3 + 2*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-e07ba370ee736991
  • 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-e07ba370ee736991
  • 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-e07ba370ee736991
  • 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-e07ba370ee736991

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