Real-rooted atlas

A207606 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(72 + 96\,n + 32\,n^{2}\right)\,P_{n}(x) = \left(144 + 276\,x + 114\,x^{2} + 192\,n + 400\,n x + 190\,n x^{2} + 15\,n x^{3} + 64\,n^{2} + 144\,n^{2} x + 76\,n^{2} x^{2} + 15\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-48\,x - 180\,x^{2} - 114\,x^{3} - 32\,n x - 120\,n x^{2} - 112\,n x^{3} - 30\,n x^{4}\right)\,P_{n-1}'(x) + \left(12\,x^{3} + 36\,x^{4} + 15\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 8 + 22\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
08221343
116688834206
23218437231089987
3644641280166810202334729
41281120392069926672321661022658
525626241113625280330402478098481597108561
65126016300168284813800014168887364295224181520147
71024135687782425312051340867209656675229635687080109462492174

All 15 rows

Technical details
Node id
n-564ce6e2f7862c99 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-99b7e9fbb1c44352

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:    8,    22,     13;
  1:   16,    68,     88,      34;
  2:   32,   184,    372,     310,      89;
  3:   64,   464,   1280,    1668,    1020,      233;
  4:  128,  1120,   3920,    6992,    6672,     3216,      610;
  5:  256,  2624,  11136,   25280,   33040,    24780,     9848,     1597;
  6:  512,  6016,  30016,   82848,  138000,   141688,    87364,    29522,    4181;
  7: 1024, 13568,  77824,  253120,  513408,   672096,   566752,   296356,   87080,   10946;
  8: 2048, 30208, 195840,  733696, 1756160,  2801568,  3011792,  2151280,  975864,  253610,  28657;
  9: 4096, 66560, 481280, 2041344, 5635584, 10618496, 13914880, 12665840, 7838816, 3138456, 731064, 75025;
%F Formula
The row polynomials satisfy (32*n^2+96*n+72)*P(n,x) = (64*n^2+192*n+144+(144*n^2+400*n+276)*x+(76*n^2+190*n+114)*x^2+(15*n^2+15*n)*x^3)*P(n-1,x) - ((32*n+48)*x+(120*n+180)*x^2+(112*n+114)*x^3+30*n*x^4)*d/dx P(n-1,x) + (12*x^3+36*x^4+15*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 8 + 22*x + 13*x^2.
%t Mathematica
P[0, x_] := 8 + 22*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64*n^2 + 192*n + 144 + (144*n^2 + 400*n + 276)*x + (76*n^2 + 190*n + 114)*x^2 + (15*n^2 + 15*n)*x^3)*P[n - 1, x] - ((32*n + 48)*x + (120*n + 180)*x^2 + (112*n + 114)*x^3 + 30*n*x^4)*D[P[n - 1, x], x] + (12*x^3 + 36*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(32*n^2 + 96*n + 72)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8 + 22*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((64*n^2 + 192*n + 144 + (144*n^2 + 400*n + 276)*x + (76*n^2 + 190*n + 114)*x^2 + (15*n^2 + 15*n)*x^3)*v[i - 1] - ((32*n + 48)*x + (120*n + 180)*x^2 + (112*n + 114)*x^3 + 30*n*x^4)*deriv(v[i - 1], x) + (12*x^3 + 36*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(32*n^2 + 96*n + 72)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023