Real-rooted atlas

A114656 → θ+1 → (θ+1)²

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-4\,n + 2\,n x + 4\,n^{2} - 14\,n^{2} x + 20\,n^{3} x - n^{3} x^{2} - 8\,n^{4} x + 3\,n^{4} x^{2} - 3\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(32\,x + 28\,x^{2} + 2\,x^{3} - 8\,n - 76\,n x - 54\,n x^{2} - 5\,n x^{3} + 8\,n^{2} + 20\,n^{2} x - 12\,n^{2} x^{2} + 40\,n^{3} x + 78\,n^{3} x^{2} + 9\,n^{3} x^{3} - 16\,n^{4} x - 36\,n^{4} x^{2} - 6\,n^{4} x^{3} - 6\,n^{5} x^{2} - 3\,n^{5} x^{3} + 2\,n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 12\,x^{2} + 6\,x^{3} + x^{4} + 16\,n x + 16\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} + 16\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{2} x^{4} - 32\,n^{3} x^{2} + 8\,n^{3} x^{3} + 4\,n^{3} x^{4} + 2\,n^{4} x^{3} - n^{4} x^{4} + 4\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
22810
34482779
4819232464588
516640216012801254221
63219201080012800375021629518
76453764536089600525009072343202315
812814336169344501760490000169344192085121364632
925636864580608240844835280002032128460992368647299084889
105129216018662401032192021168000182891526914880110592065610100059825394
111024225280570240040550400110880000134120448760636802027520024057001100001331390335463
1220485406721672704014868480052272000084304281666936038426763264052925400484000017569217282526653220
13409612779524744396851544064022651200004696952832497239142427833794568256362401258400009135984269568219716242894357
148192298188813083033617056399369161152000237457059843232054425624122621952100177197122290288000277124848163537923998542744103791373494
15163846881280352235520542703616034978944000110813294592188536508160180919664640100177197120320640320005819621808572382720279897805762403375659696383779
Technical details
Node id
n-570a472e1ba455c2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-64a0e8e2d31a6bd3

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 -1)
  • 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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   2,     8;
   3:   4,    48,      27;
   4:   8,   192,     324,       64;
   5:  16,   640,    2160,     1280,      125;
   6:  32,  1920,   10800,    12800,     3750,      216;
   7:  64,  5376,   45360,    89600,    52500,     9072,     343;
   8: 128, 14336,  169344,   501760,   490000,   169344,   19208,     512;
   9: 256, 36864,  580608,  2408448,  3528000,  2032128,  460992,   36864,   729;
  10: 512, 92160, 1866240, 10321920, 21168000, 18289152, 6914880, 1105920, 65610, 1000;
%F Formula
The row polynomials satisfy (4*n^2-4*n-(8*n^4-20*n^3+14*n^2-2*n)*x+(n^6-3*n^5+3*n^4-n^3)*x^2)*P(n,x) = (8*n^2-8*n-(16*n^4-40*n^3-20*n^2+76*n-32)*x+(2*n^6-6*n^5-36*n^4+78*n^3-12*n^2-54*n+28)*x^2+(3*n^6-3*n^5-6*n^4+9*n^3-5*n+2)*x^3)*P(n-1,x) + ((16*n-8)*x-(32*n^3-16*n^2-16*n+12)*x^2+(4*n^5+2*n^4+8*n^3-6*n^2-8*n+6)*x^3-(2*n^5+n^4-4*n^3+n^2+2*n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 8*n - (16*n^4 - 40*n^3 - 20*n^2 + 76*n - 32)*x + (2*n^6 - 6*n^5 - 36*n^4 + 78*n^3 - 12*n^2 - 54*n + 28)*x^2 + (3*n^6 - 3*n^5 - 6*n^4 + 9*n^3 - 5*n + 2)*x^3)*P[n - 1, x] + ((16*n - 8)*x - (32*n^3 - 16*n^2 - 16*n + 12)*x^2 + (4*n^5 + 2*n^4 + 8*n^3 - 6*n^2 - 8*n + 6)*x^3 - (2*n^5 + n^4 - 4*n^3 + n^2 + 2*n - 1)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 4*n - (8*n^4 - 20*n^3 + 14*n^2 - 2*n)*x + (n^6 - 3*n^5 + 3*n^4 - n^3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^2 - 8*n - (16*n^4 - 40*n^3 - 20*n^2 + 76*n - 32)*x + (2*n^6 - 6*n^5 - 36*n^4 + 78*n^3 - 12*n^2 - 54*n + 28)*x^2 + (3*n^6 - 3*n^5 - 6*n^4 + 9*n^3 - 5*n + 2)*x^3)*v[i - 1] + ((16*n - 8)*x - (32*n^3 - 16*n^2 - 16*n + 12)*x^2 + (4*n^5 + 2*n^4 + 8*n^3 - 6*n^2 - 8*n + 6)*x^3 - (2*n^5 + n^4 - 4*n^3 + n^2 + 2*n - 1)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 4*n - (8*n^4 - 20*n^3 + 14*n^2 - 2*n)*x + (n^6 - 3*n^5 + 3*n^4 - 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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023