Real-rooted atlas

A091044 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x + 16\,n^{2} - 4\,n^{2} x + 16\,n^{3} + 12\,n^{3} x + n^{3} x^{2} + 24\,n^{4} x + 4\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-16\,x - 2\,x^{2} - x^{3} + 16\,n - 24\,n x - 44\,n x^{2} - 5\,n x^{3} + 64\,n^{2} + 112\,n^{2} x - 2\,n^{2} x^{2} - 5\,n^{2} x^{3} + 64\,n^{3} + 240\,n^{3} x + 172\,n^{3} x^{2} + 31\,n^{3} x^{3} + 96\,n^{4} x + 136\,n^{4} x^{2} + 40\,n^{4} x^{3} + 16\,n^{5} x^{2} + 12\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 12\,x^{2} - 6\,x^{3} - 2\,x^{4} - 32\,n x - 80\,n x^{2} - 62\,n x^{3} - 14\,n x^{4} - 32\,n^{2} x - 128\,n^{2} x^{2} - 134\,n^{2} x^{3} - 38\,n^{2} x^{4} - 48\,n^{3} x^{2} - 80\,n^{3} x^{3} - 32\,n^{3} x^{4} - 8\,n^{4} x^{3} - 8\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2166427107
36438436064872
42562048302412801256733
51024102402073614336350021650052
6409649152126720122880504008064343361655
716384229376718848901120528000145152164645122555856
865536104857638707205963776457600018247683622083072072917743033
92621444718592200540163670016034944000184504325381376811008534601000121376188
1010485762097152010085990421390950424371200016102195262782720140574721667952880001331820120931
1141943049227468849545216011953766401587609600126340300862097817618743296033359040320320013842417285483423928
1216777216402653184238760755264592281609805824000914463129654455009282085617664504092160732160005813808209664219736331173829
1367108864174483046411324620800339570851845803212800062129700864435640074242032140288062690734081244672000150605312100638723075802744238815609396
14268435456751619276852999225344174483046400331612160000401118068736324167467008178828345344675130982401719910400028615009282932899841673235243904033751558877108975
Technical details
Node id
n-20218a6390dee769 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f7c15ce40ae0ded9

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      4,       8;
  2:     16,      64,       27;
  3:     64,     384,      360,       64;
  4:    256,    2048,     3024,     1280,      125;
  5:   1024,   10240,    20736,    14336,     3500,      216;
  6:   4096,   49152,   126720,   122880,    50400,     8064,     343;
  7:  16384,  229376,   718848,   901120,   528000,   145152,   16464,    512;
  8:  65536, 1048576,  3870720,  5963776,  4576000,  1824768,  362208,  30720,   729;
  9: 262144, 4718592, 20054016, 36700160, 34944000, 18450432, 5381376, 811008, 53460, 1000;
%F Formula
The row polynomials satisfy (16*n^3+16*n^2+4*n+(24*n^4+12*n^3-4*n^2-2*n)*x+(4*n^5+4*n^4+n^3)*x^2)*P(n,x) = (64*n^3+64*n^2+16*n+(96*n^4+240*n^3+112*n^2-24*n-16)*x+(16*n^5+136*n^4+172*n^3-2*n^2-44*n-2)*x^2+(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^3)*P(n-1,x) - ((32*n^2+32*n+8)*x+(48*n^3+128*n^2+80*n+12)*x^2+(8*n^4+80*n^3+134*n^2+62*n+6)*x^3+(8*n^4+32*n^3+38*n^2+14*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[((64*n^3 + 64*n^2 + 16*n + (96*n^4 + 240*n^3 + 112*n^2 - 24*n - 16)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*P[n - 1, x] - ((32*n^2 + 32*n + 8)*x + (48*n^3 + 128*n^2 + 80*n + 12)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*D[P[n - 1, x], x])/(16*n^3 + 16*n^2 + 4*n + (24*n^4 + 12*n^3 - 4*n^2 - 2*n)*x + (4*n^5 + 4*n^4 + 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] = ((64*n^3 + 64*n^2 + 16*n + (96*n^4 + 240*n^3 + 112*n^2 - 24*n - 16)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*v[i - 1] - ((32*n^2 + 32*n + 8)*x + (48*n^3 + 128*n^2 + 80*n + 12)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*deriv(v[i - 1], x))/(16*n^3 + 16*n^2 + 4*n + (24*n^4 + 12*n^3 - 4*n^2 - 2*n)*x + (4*n^5 + 4*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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