Real-rooted atlas

A172431 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(40\,n + 56\,n x + 80\,n^{2} x + 32\,n^{2} x^{2} + 64\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-40 - 176\,x - 224\,x^{2} - 64\,x^{3} + 160\,n + 504\,n x + 320\,n x^{2} - 128\,n x^{3} + 400\,n^{2} x + 1024\,n^{2} x^{2} + 576\,n^{2} x^{3} + 256\,n^{3} x^{2} + 384\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 160\,x - 372\,x^{2} - 368\,x^{3} - 128\,x^{4} - 100\,n x - 520\,n x^{2} - 896\,n x^{3} - 512\,n x^{4} - 80\,n^{2} x^{2} - 288\,n^{2} x^{3} - 256\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
28403684
321152264128565
455520133212804003587
51441680568881925200115222056
63775232220324236840880188163136132841
79871588079992192640250400177408627208192788219

All 15 rows

Technical details
Node id
n-876faf6eaa9af2a5 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1691e7df67786cdc

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:    3,      8;
  2:    8,     40,     36;
  3:   21,    152,    264,     128;
  4:   55,    520,   1332,    1280,     400;
  5:  144,   1680,   5688,    8192,    5200,    1152;
  6:  377,   5232,  22032,   42368,   40880,   18816,    3136;
  7:  987,  15880,  79992,  192640,  250400,  177408,   62720,    8192;
  8: 2584,  47288, 277308,  802816, 1314800, 1264896,  696192,  196608,  20736;
  9: 6765, 138760, 928536, 3141120, 6211600, 7570560, 5698560, 2531328, 587520, 51200;
%F Formula
The row polynomials satisfy (40*n+(80*n^2+56*n)*x+(64*n^3+32*n^2)*x^2)*P(n,x) = (160*n-40+(400*n^2+504*n-176)*x+(256*n^3+1024*n^2+320*n-224)*x^2+(384*n^3+576*n^2-128*n-64)*x^3)*P(n-1,x) - (25+(100*n+160)*x+(80*n^2+520*n+372)*x^2+(288*n^2+896*n+368)*x^3+(256*n^2+512*n+128)*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[((160*n - 40 + (400*n^2 + 504*n - 176)*x + (256*n^3 + 1024*n^2 + 320*n - 224)*x^2 + (384*n^3 + 576*n^2 - 128*n - 64)*x^3)*P[n - 1, x] - (25 + (100*n + 160)*x + (80*n^2 + 520*n + 372)*x^2 + (288*n^2 + 896*n + 368)*x^3 + (256*n^2 + 512*n + 128)*x^4)*D[P[n - 1, x], x])/(40*n + (80*n^2 + 56*n)*x + (64*n^3 + 32*n^2)*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] = ((160*n - 40 + (400*n^2 + 504*n - 176)*x + (256*n^3 + 1024*n^2 + 320*n - 224)*x^2 + (384*n^3 + 576*n^2 - 128*n - 64)*x^3)*v[i - 1] - (25 + (100*n + 160)*x + (80*n^2 + 520*n + 372)*x^2 + (288*n^2 + 896*n + 368)*x^3 + (256*n^2 + 512*n + 128)*x^4)*deriv(v[i - 1], x))/(40*n + (80*n^2 + 56*n)*x + (64*n^3 + 32*n^2)*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