Real-rooted atlas

A091044 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 6\,n x + 16\,n^{2} - 12\,n^{2} x + 16\,n^{3} + 36\,n^{3} x + 9\,n^{3} x^{2} + 72\,n^{4} x + 36\,n^{4} x^{2} + 36\,n^{5} x^{2}\right)\,P_{n}(x) = \left(-48\,x - 18\,x^{2} - 27\,x^{3} + 16\,n - 72\,n x - 396\,n x^{2} - 135\,n x^{3} + 64\,n^{2} + 336\,n^{2} x - 18\,n^{2} x^{2} - 135\,n^{2} x^{3} + 64\,n^{3} + 720\,n^{3} x + 1548\,n^{3} x^{2} + 837\,n^{3} x^{3} + 288\,n^{4} x + 1224\,n^{4} x^{2} + 1080\,n^{4} x^{3} + 144\,n^{5} x^{2} + 324\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-8\,x - 36\,x^{2} - 54\,x^{3} - 54\,x^{4} - 32\,n x - 240\,n x^{2} - 558\,n x^{3} - 378\,n x^{4} - 32\,n^{2} x - 384\,n^{2} x^{2} - 1206\,n^{2} x^{3} - 1026\,n^{2} x^{4} - 144\,n^{3} x^{2} - 720\,n^{3} x^{3} - 864\,n^{3} x^{4} - 72\,n^{4} x^{3} - 216\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
142428
216192243451
3641152324017286184
4256614427216345601012578301
510243072018662438707228350052488941428
64096147456114048033177604082400195955225004710901791
7163846881286469632243302404276800035271936120022561119744122666320

All 15 rows

Technical details
Node id
n-a36b339e55dd2f9e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-917b1bbf533fd361

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,       24;
  2:     16,      192,       243;
  3:     64,     1152,      3240,      1728;
  4:    256,     6144,     27216,     34560,      10125;
  5:   1024,    30720,    186624,    387072,     283500,      52488;
  6:   4096,   147456,   1140480,   3317760,    4082400,    1959552,     250047;
  7:  16384,   688128,   6469632,  24330240,   42768000,   35271936,   12002256,    1119744;
  8:  65536,  3145728,  34836480, 161021952,  370656000,  443418624,  264049632,   67184640,   4782969;
  9: 262144, 14155776, 180486144, 990904320, 2830464000, 4483454976, 3923023104, 1773674496, 350751060, 19683000;
%F Formula
The row polynomials satisfy (16*n^3+16*n^2+4*n+(72*n^4+36*n^3-12*n^2-6*n)*x+(36*n^5+36*n^4+9*n^3)*x^2)*P(n,x) = (64*n^3+64*n^2+16*n+(288*n^4+720*n^3+336*n^2-72*n-48)*x+(144*n^5+1224*n^4+1548*n^3-18*n^2-396*n-18)*x^2+(324*n^5+1080*n^4+837*n^3-135*n^2-135*n-27)*x^3)*P(n-1,x) - ((32*n^2+32*n+8)*x+(144*n^3+384*n^2+240*n+36)*x^2+(72*n^4+720*n^3+1206*n^2+558*n+54)*x^3+(216*n^4+864*n^3+1026*n^2+378*n+54)*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 + (288*n^4 + 720*n^3 + 336*n^2 - 72*n - 48)*x + (144*n^5 + 1224*n^4 + 1548*n^3 - 18*n^2 - 396*n - 18)*x^2 + (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^3)*P[n - 1, x] - ((32*n^2 + 32*n + 8)*x + (144*n^3 + 384*n^2 + 240*n + 36)*x^2 + (72*n^4 + 720*n^3 + 1206*n^2 + 558*n + 54)*x^3 + (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^4)*D[P[n - 1, x], x])/(16*n^3 + 16*n^2 + 4*n + (72*n^4 + 36*n^3 - 12*n^2 - 6*n)*x + (36*n^5 + 36*n^4 + 9*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 + (288*n^4 + 720*n^3 + 336*n^2 - 72*n - 48)*x + (144*n^5 + 1224*n^4 + 1548*n^3 - 18*n^2 - 396*n - 18)*x^2 + (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^3)*v[i - 1] - ((32*n^2 + 32*n + 8)*x + (144*n^3 + 384*n^2 + 240*n + 36)*x^2 + (72*n^4 + 720*n^3 + 1206*n^2 + 558*n + 54)*x^3 + (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^4)*deriv(v[i - 1], x))/(16*n^3 + 16*n^2 + 4*n + (72*n^4 + 36*n^3 - 12*n^2 - 6*n)*x + (36*n^5 + 36*n^4 + 9*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