Real-rooted atlas

A062264 → (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
\(\left(4\,n + 16\,n x + n^{2} - 52\,n^{2} x - 30\,n^{3} x + 64\,n^{3} x^{2} - 4\,n^{4} x + 48\,n^{4} x^{2} + 12\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(48\,x - 216\,x^{2} + 192\,x^{3} + 4\,n + 56\,n x - 552\,n x^{2} + 640\,n x^{3} + n^{2} - 40\,n^{2} x - 441\,n^{2} x^{2} + 840\,n^{2} x^{3} - 30\,n^{3} x - 106\,n^{3} x^{2} + 564\,n^{3} x^{3} - 4\,n^{4} x + 27\,n^{4} x^{2} + 219\,n^{4} x^{3} + 12\,n^{5} x^{2} + 42\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(4\,x - 48\,x^{2} + 108\,x^{3} - 64\,x^{4} + 2\,n x - 96\,n x^{2} + 286\,n x^{3} - 192\,n x^{4} - 56\,n^{2} x^{2} + 272\,n^{2} x^{3} - 216\,n^{2} x^{4} - 8\,n^{3} x^{2} + 124\,n^{3} x^{3} - 116\,n^{3} x^{4} + 26\,n^{4} x^{3} - 26\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112021
2148135184
31845675601212
411281512358417506975
5118032401344015750453637147
61240607538400787505443210290188188
71308103959240028875034927215846621120920712

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-bd58d17a1a7a7190 · revision 1 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9fad260d5e548d70

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,  20;
  2: 1,  48,   135;
  3: 1,  84,   567,    560;
  4: 1, 128,  1512,   3584,    1750;
  5: 1, 180,  3240,  13440,   15750,    4536;
  6: 1, 240,  6075,  38400,   78750,   54432,   10290;
  7: 1, 308, 10395,  92400,  288750,  349272,  158466,   21120;
  8: 1, 384, 16632, 197120,  866250, 1596672, 1267728,  405504,  40095;
  9: 1, 468, 25272, 384384, 2252250, 5837832, 7063056, 3953664, 938223, 71500;
%F Formula
The row polynomials satisfy (n^2+4*n-(4*n^4+30*n^3+52*n^2-16*n)*x+(n^6+12*n^5+48*n^4+64*n^3)*x^2)*P(n,x) = (n^2+4*n-(4*n^4+30*n^3+40*n^2-56*n-48)*x+(n^6+12*n^5+27*n^4-106*n^3-441*n^2-552*n-216)*x^2+(3*n^6+42*n^5+219*n^4+564*n^3+840*n^2+640*n+192)*x^3)*P(n-1,x) + ((2*n+4)*x-(8*n^3+56*n^2+96*n+48)*x^2+(2*n^5+26*n^4+124*n^3+272*n^2+286*n+108)*x^3-(2*n^5+26*n^4+116*n^3+216*n^2+192*n+64)*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[((n^2 + 4*n - (4*n^4 + 30*n^3 + 40*n^2 - 56*n - 48)*x + (n^6 + 12*n^5 + 27*n^4 - 106*n^3 - 441*n^2 - 552*n - 216)*x^2 + (3*n^6 + 42*n^5 + 219*n^4 + 564*n^3 + 840*n^2 + 640*n + 192)*x^3)*P[n - 1, x] + ((2*n + 4)*x - (8*n^3 + 56*n^2 + 96*n + 48)*x^2 + (2*n^5 + 26*n^4 + 124*n^3 + 272*n^2 + 286*n + 108)*x^3 - (2*n^5 + 26*n^4 + 116*n^3 + 216*n^2 + 192*n + 64)*x^4)*D[P[n - 1, x], x])/(n^2 + 4*n - (4*n^4 + 30*n^3 + 52*n^2 - 16*n)*x + (n^6 + 12*n^5 + 48*n^4 + 64*n^3)*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] = ((n^2 + 4*n - (4*n^4 + 30*n^3 + 40*n^2 - 56*n - 48)*x + (n^6 + 12*n^5 + 27*n^4 - 106*n^3 - 441*n^2 - 552*n - 216)*x^2 + (3*n^6 + 42*n^5 + 219*n^4 + 564*n^3 + 840*n^2 + 640*n + 192)*x^3)*v[i - 1] + ((2*n + 4)*x - (8*n^3 + 56*n^2 + 96*n + 48)*x^2 + (2*n^5 + 26*n^4 + 124*n^3 + 272*n^2 + 286*n + 108)*x^3 - (2*n^5 + 26*n^4 + 116*n^3 + 216*n^2 + 192*n + 64)*x^4)*deriv(v[i - 1], x))/(n^2 + 4*n - (4*n^4 + 30*n^3 + 52*n^2 - 16*n)*x + (n^6 + 12*n^5 + 48*n^4 + 64*n^3)*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-4f5eaf3b19bd0a90
  • 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-4f5eaf3b19bd0a90
  • 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-4f5eaf3b19bd0a90
  • 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-4f5eaf3b19bd0a90

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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.