Real-rooted atlas

A123519 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12 + 18\,x + 14\,n + 33\,n x + 4\,n^{2} + 20\,n^{2} x + 4\,n^{3} x\right)\,P_{n}(x) = \left(24 + 92\,x + 73\,x^{2} + 28\,n + 142\,n x + 131\,n x^{2} + 8\,n^{2} + 64\,n^{2} x + 72\,n^{2} x^{2} + 8\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 39\,x^{2} - 46\,x^{3} - 4\,n x - 28\,n x^{2} - 40\,n x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 4 + 12\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0412319
184036488
216112180805393
33228867256015061708
46470421602688140025277275
5128166463361056084003024392830512
625638401747236608396002217658805769126417
7512870446080116480160160123552517441056081010518612

All 15 rows

Technical details
Node id
n-790e61cd1a4f1614 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d88a651a3889a86b

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:    4,    12,      3;
  1:    8,    40,     36,      4;
  2:   16,   112,    180,     80,       5;
  3:   32,   288,    672,    560,     150,       6;
  4:   64,   704,   2160,   2688,    1400,     252,       7;
  5:  128,  1664,   6336,  10560,    8400,    3024,     392,      8;
  6:  256,  3840,  17472,  36608,   39600,   22176,    5880,    576,      9;
  7:  512,  8704,  46080, 116480,  160160,  123552,   51744,  10560,    810,    10;
  8: 1024, 19456, 117504, 348160,  582400,  576576,  336336, 109824,  17820,  1100,   11;
  9: 2048, 43008, 291840, 992256, 1958400, 2376192, 1793792, 823680, 216216, 28600, 1452, 12;
%F Formula
The row polynomials satisfy (4*n^2+14*n+12+(4*n^3+20*n^2+33*n+18)*x)*P(n,x) = (8*n^2+28*n+24+(8*n^3+64*n^2+142*n+92)*x+(12*n^3+72*n^2+131*n+73)*x^2)*P(n-1,x) - ((4*n+8)*x+(4*n^2+28*n+39)*x^2+(8*n^2+40*n+46)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 12*x + 3*x^2.
%t Mathematica
P[0, x_] := 4 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 28*n + 24 + (8*n^3 + 64*n^2 + 142*n + 92)*x + (12*n^3 + 72*n^2 + 131*n + 73)*x^2)*P[n - 1, x] - ((4*n + 8)*x + (4*n^2 + 28*n + 39)*x^2 + (8*n^2 + 40*n + 46)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12 + (4*n^3 + 20*n^2 + 33*n + 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 12*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 28*n + 24 + (8*n^3 + 64*n^2 + 142*n + 92)*x + (12*n^3 + 72*n^2 + 131*n + 73)*x^2)*v[i - 1] - ((4*n + 8)*x + (4*n^2 + 28*n + 39)*x^2 + (8*n^2 + 40*n + 46)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12 + (4*n^3 + 20*n^2 + 33*n + 18)*x)); 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

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